Fourier transform - Wikipedia

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<div class="vector-toc-text"> 1.4 Extension of the definition </div> </a> <ul id="toc-Extension_of_the_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> 2 Background </div> </a> <button aria-controls="toc-Background-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Background subsection </button> <ul id="toc-Background-sublist" class="vector-toc-list"> <li id="toc-History" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2.1 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_sinusoids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_sinusoids"> <div class="vector-toc-text"> 2.2 Complex sinusoids </div> </a> <ul id="toc-Complex_sinusoids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negative_frequency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_frequency"> <div class="vector-toc-text"> 2.3 Negative frequency </div> </a> <ul id="toc-Negative_frequency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_transform_for_periodic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_transform_for_periodic_functions"> <div class="vector-toc-text"> 2.4 Fourier transform for periodic functions </div> </a> <ul id="toc-Fourier_transform_for_periodic_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sampling_the_Fourier_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sampling_the_Fourier_transform"> <div class="vector-toc-text"> 2.5 Sampling the Fourier transform </div> </a> <ul id="toc-Sampling_the_Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 3 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_the_Fourier_transform" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_of_the_Fourier_transform"> <div class="vector-toc-text"> 4 Properties of the Fourier transform </div> </a> <button aria-controls="toc-Properties_of_the_Fourier_transform-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties of the Fourier transform subsection </button> <ul id="toc-Properties_of_the_Fourier_transform-sublist" class="vector-toc-list"> <li id="toc-Basic_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_properties"> <div class="vector-toc-text"> 4.1 Basic properties </div> </a> <ul id="toc-Basic_properties-sublist" class="vector-toc-list"> <li id="toc-Linearity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Linearity"> <div class="vector-toc-text"> 4.1.1 Linearity </div> </a> <ul id="toc-Linearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_shifting" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time_shifting"> <div class="vector-toc-text"> 4.1.2 Time shifting </div> </a> <ul id="toc-Time_shifting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frequency_shifting" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Frequency_shifting"> <div class="vector-toc-text"> 4.1.3 Frequency shifting </div> </a> <ul id="toc-Frequency_shifting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_scaling" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time_scaling"> <div class="vector-toc-text"> 4.1.4 Time scaling </div> </a> <ul id="toc-Time_scaling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> 4.1.5 Symmetry </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjugation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conjugation"> <div class="vector-toc-text"> 4.1.6 Conjugation </div> </a> <ul id="toc-Conjugation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_and_imaginary_part_in_time" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Real_and_imaginary_part_in_time"> <div class="vector-toc-text"> 4.1.7 Real and imaginary part in time </div> </a> <ul id="toc-Real_and_imaginary_part_in_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zero_frequency_component" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zero_frequency_component"> <div class="vector-toc-text"> 4.1.8 Zero frequency component </div> </a> <ul id="toc-Zero_frequency_component-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Invertibility_and_periodicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invertibility_and_periodicity"> <div class="vector-toc-text"> 4.2 Invertibility and periodicity </div> </a> <ul id="toc-Invertibility_and_periodicity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> 4.3 Units </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_continuity_and_the_Riemann–Lebesgue_lemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_continuity_and_the_Riemann–Lebesgue_lemma"> <div class="vector-toc-text"> 4.4 Uniform continuity and the Riemann–Lebesgue lemma </div> </a> <ul id="toc-Uniform_continuity_and_the_Riemann–Lebesgue_lemma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plancherel_theorem_and_Parseval's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plancherel_theorem_and_Parseval's_theorem"> <div class="vector-toc-text"> 4.5 Plancherel theorem and Parseval's theorem </div> </a> <ul id="toc-Plancherel_theorem_and_Parseval's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_summation_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_summation_formula"> <div class="vector-toc-text"> 4.6 Poisson summation formula </div> </a> <ul id="toc-Poisson_summation_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differentiation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differentiation"> <div class="vector-toc-text"> 4.7 Differentiation </div> </a> <ul id="toc-Differentiation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convolution_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convolution_theorem"> <div class="vector-toc-text"> 4.8 Convolution theorem </div> </a> <ul id="toc-Convolution_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cross-correlation_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cross-correlation_theorem"> <div class="vector-toc-text"> 4.9 Cross-correlation theorem </div> </a> <ul id="toc-Cross-correlation_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenfunctions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenfunctions"> <div class="vector-toc-text"> 4.10 Eigenfunctions </div> </a> <ul id="toc-Eigenfunctions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_the_Heisenberg_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_the_Heisenberg_group"> <div class="vector-toc-text"> 4.11 Connection with the Heisenberg group </div> </a> <ul id="toc-Connection_with_the_Heisenberg_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_domain" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_domain"> <div class="vector-toc-text"> 5 Complex domain </div> </a> <button aria-controls="toc-Complex_domain-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Complex domain subsection </button> <ul id="toc-Complex_domain-sublist" class="vector-toc-list"> <li id="toc-Laplace_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace_transform"> <div class="vector-toc-text"> 5.1 Laplace transform </div> </a> <ul id="toc-Laplace_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inversion"> <div class="vector-toc-text"> 5.2 Inversion </div> </a> <ul id="toc-Inversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fourier_transform_on_Euclidean_space" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier_transform_on_Euclidean_space"> <div class="vector-toc-text"> 6 Fourier transform on Euclidean space </div> </a> <button aria-controls="toc-Fourier_transform_on_Euclidean_space-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Fourier transform on Euclidean space subsection </button> <ul id="toc-Fourier_transform_on_Euclidean_space-sublist" class="vector-toc-list"> <li id="toc-Uncertainty_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncertainty_principle"> <div class="vector-toc-text"> 6.1 Uncertainty principle </div> </a> <ul id="toc-Uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sine_and_cosine_transforms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sine_and_cosine_transforms"> <div class="vector-toc-text"> 6.2 Sine and cosine transforms </div> </a> <ul id="toc-Sine_and_cosine_transforms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_harmonics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spherical_harmonics"> <div class="vector-toc-text"> 6.3 Spherical harmonics </div> </a> <ul id="toc-Spherical_harmonics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Restriction_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Restriction_problems"> <div class="vector-toc-text"> 6.4 Restriction problems </div> </a> <ul id="toc-Restriction_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fourier_transform_on_function_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fourier_transform_on_function_spaces"> <div class="vector-toc-text"> 7 Fourier transform on function spaces </div> </a> <button aria-controls="toc-Fourier_transform_on_function_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Fourier transform on function spaces subsection </button> <ul id="toc-Fourier_transform_on_function_spaces-sublist" class="vector-toc-list"> <li id="toc-On_Lp_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#On_Lp_spaces"> <div class="vector-toc-text"> 7.1 On Lp spaces </div> </a> <ul id="toc-On_Lp_spaces-sublist" class="vector-toc-list"> <li id="toc-On_L1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#On_L1"> <div class="vector-toc-text"> 7.1.1 On L1 </div> </a> <ul id="toc-On_L1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-On_L2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#On_L2"> <div class="vector-toc-text"> 7.1.2 On L2 </div> </a> <ul id="toc-On_L2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-On_other_Lp" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#On_other_Lp"> <div class="vector-toc-text"> 7.1.3 On other Lp </div> </a> <ul id="toc-On_other_Lp-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tempered_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tempered_distributions"> <div class="vector-toc-text"> 7.2 Tempered distributions </div> </a> <ul id="toc-Tempered_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 8 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Fourier–Stieltjes_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier–Stieltjes_transform"> <div class="vector-toc-text"> 8.1 Fourier–Stieltjes transform </div> </a> <ul id="toc-Fourier–Stieltjes_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Locally_compact_abelian_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Locally_compact_abelian_groups"> <div class="vector-toc-text"> 8.2 Locally compact abelian groups </div> </a> <ul id="toc-Locally_compact_abelian_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gelfand_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gelfand_transform"> <div class="vector-toc-text"> 8.3 Gelfand transform </div> </a> <ul id="toc-Gelfand_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compact_non-abelian_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compact_non-abelian_groups"> <div class="vector-toc-text"> 8.4 Compact non-abelian groups </div> </a> <ul id="toc-Compact_non-abelian_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Alternatives" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Alternatives"> <div class="vector-toc-text"> 9 Alternatives </div> </a> <ul id="toc-Alternatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 10 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Analysis_of_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analysis_of_differential_equations"> <div class="vector-toc-text"> 10.1 Analysis of differential equations </div> </a> <ul id="toc-Analysis_of_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier-transform_spectroscopy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier-transform_spectroscopy"> <div class="vector-toc-text"> 10.2 Fourier-transform spectroscopy </div> </a> <ul id="toc-Fourier-transform_spectroscopy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 10.3 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signal_processing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signal_processing"> <div class="vector-toc-text"> 10.4 Signal processing </div> </a> <ul id="toc-Signal_processing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_notations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_notations"> <div class="vector-toc-text"> 11 Other notations </div> </a> <ul id="toc-Other_notations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation_methods" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computation_methods"> <div class="vector-toc-text"> 12 Computation methods </div> </a> <button aria-controls="toc-Computation_methods-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Computation methods subsection </button> <ul id="toc-Computation_methods-sublist" class="vector-toc-list"> <li id="toc-Discrete_Fourier_transforms_and_fast_Fourier_transforms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_Fourier_transforms_and_fast_Fourier_transforms"> <div class="vector-toc-text"> 12.1 Discrete Fourier transforms and fast Fourier transforms </div> </a> <ul id="toc-Discrete_Fourier_transforms_and_fast_Fourier_transforms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytic_integration_of_closed-form_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytic_integration_of_closed-form_functions"> <div class="vector-toc-text"> 12.2 Analytic integration of closed-form functions </div> </a> <ul id="toc-Analytic_integration_of_closed-form_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_integration_of_closed-form_continuous_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_integration_of_closed-form_continuous_functions"> <div class="vector-toc-text"> 12.3 Numerical integration of closed-form continuous functions </div> </a> <ul id="toc-Numerical_integration_of_closed-form_continuous_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_integration_of_a_series_of_ordered_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_integration_of_a_series_of_ordered_pairs"> <div class="vector-toc-text"> 12.4 Numerical integration of a series of ordered pairs </div> </a> <ul id="toc-Numerical_integration_of_a_series_of_ordered_pairs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tables_of_important_Fourier_transforms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tables_of_important_Fourier_transforms"> <div class="vector-toc-text"> 13 Tables of important Fourier transforms </div> </a> <button aria-controls="toc-Tables_of_important_Fourier_transforms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Tables of important Fourier transforms subsection </button> <ul id="toc-Tables_of_important_Fourier_transforms-sublist" class="vector-toc-list"> <li id="toc-Functional_relationships,_one-dimensional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functional_relationships,_one-dimensional"> <div class="vector-toc-text"> 13.1 Functional relationships, one-dimensional </div> </a> <ul id="toc-Functional_relationships,_one-dimensional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square-integrable_functions,_one-dimensional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square-integrable_functions,_one-dimensional"> <div class="vector-toc-text"> 13.2 Square-integrable functions, one-dimensional </div> </a> <ul id="toc-Square-integrable_functions,_one-dimensional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributions,_one-dimensional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributions,_one-dimensional"> <div class="vector-toc-text"> 13.3 Distributions, one-dimensional </div> </a> <ul id="toc-Distributions,_one-dimensional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-dimensional_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-dimensional_functions"> <div class="vector-toc-text"> 13.4 Two-dimensional functions </div> </a> <ul id="toc-Two-dimensional_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formulas_for_general_n-dimensional_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formulas_for_general_n-dimensional_functions"> <div class="vector-toc-text"> 13.5 Formulas for general n-dimensional functions </div> </a> <ul id="toc-Formulas_for_general_n-dimensional_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 14 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 15 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 16 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 17 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 18 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Fourier transform</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" 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Available in 64 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" aria-hidden="true" > 64 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8D%8E%E1%88%AA%E1%8B%A8%E1%88%AD_%E1%88%BD%E1%8C%8D%E1%8C%8D%E1%88%AD" title="የፎሪየር ሽግግር – Amharic" lang="am" hreflang="am" data-title="የፎሪየር ሽግግር" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D9%81%D9%88%D8%B1%D9%8A%D9%8A%D9%87" title="تحويل فورييه – Arabic" lang="ar" hreflang="ar" data-title="تحويل فورييه" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tresformada_de_Fourier" title="Tresformada de Fourier – Asturian" lang="ast" hreflang="ast" data-title="Tresformada de Fourier" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Furye_%C3%A7evrilm%C9%99si" title="Furye çevrilməsi – Azerbaijani" lang="az" hreflang="az" data-title="Furye çevrilməsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A7%81%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%87_%E0%A6%B0%E0%A7%82%E0%A6%AA%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0" title="ফুরিয়ে রূপান্তর – Bangla" lang="bn" hreflang="bn" data-title="ফুরিয়ে রূপান্তর" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Fourier_pi%C3%A0n-%C5%8Da%E2%81%BF" title="Fourier piàn-ōaⁿ – Minnan" lang="nan" hreflang="nan" data-title="Fourier piàn-ōaⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B0%D1%9E%D1%82%D0%B2%D0%B0%D1%80%D1%8D%D0%BD%D0%BD%D0%B5_%D0%A4%D1%83%D1%80%E2%80%99%D0%B5" title="Пераўтварэнне Фур’е – Belarusian" lang="be" hreflang="be" data-title="Пераўтварэнне Фур’е" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B0%D1%9E%D1%82%D0%B2%D0%B0%D1%80%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5_%D0%A4%D1%83%D1%80%E2%80%99%D0%B5" title="Пераўтварэньне Фур’е – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Пераўтварэньне Фур’е" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%A4%D1%83%D1%80%D0%B8%D0%B5" title="Преобразование на Фурие – Bulgarian" lang="bg" hreflang="bg" data-title="Преобразование на Фурие" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Fouriertransformation" title="Fouriertransformation – Bavarian" lang="bar" hreflang="bar" data-title="Fouriertransformation" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier – Catalan" lang="ca" hreflang="ca" data-title="Transformada de Fourier" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fourierova_transformace" title="Fourierova transformace – Czech" lang="cs" hreflang="cs" data-title="Fourierova transformace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fouriertransformation" title="Fouriertransformation – Danish" lang="da" hreflang="da" data-title="Fouriertransformation" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fourier-Transformation" title="Fourier-Transformation – German" lang="de" hreflang="de" data-title="Fourier-Transformation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Fourier%27_teisendus" title="Fourier' teisendus – Estonian" lang="et" hreflang="et" data-title="Fourier' teisendus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CE%B1%CF%83%CF%87%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CE%A6%CE%BF%CF%85%CF%81%CE%B9%CE%AD" title="Μετασχηματισμός Φουριέ – Greek" lang="el" hreflang="el" data-title="Μετασχηματισμός Φουριέ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier – Spanish" lang="es" hreflang="es" data-title="Transformada de Fourier" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Furiera_transformo" title="Furiera transformo – Esperanto" lang="eo" hreflang="eo" data-title="Furiera transformo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fourierren_transformatu" title="Fourierren transformatu – Basque" lang="eu" hreflang="eu" data-title="Fourierren transformatu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D9%81%D9%88%D8%B1%DB%8C%D9%87" title="تبدیل فوریه – Persian" lang="fa" hreflang="fa" data-title="تبدیل فوریه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transformation_de_Fourier" title="Transformation de Fourier – French" lang="fr" hreflang="fr" data-title="Transformation de Fourier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier – Galician" lang="gl" hreflang="gl" data-title="Transformada de Fourier" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%91%B8%EB%A6%AC%EC%97%90_%EB%B3%80%ED%99%98" title="푸리에 변환 – Korean" lang="ko" hreflang="ko" data-title="푸리에 변환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A5%87_%E0%A4%B0%E0%A5%82%E0%A4%AA%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0" title="फूर्ये रूपान्तर – Hindi" lang="hi" hreflang="hi" data-title="फूर्ये रूपान्तर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Fourierova_transformacija" title="Fourierova transformacija – Croatian" lang="hr" hreflang="hr" data-title="Fourierova transformacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Transformasi_Fourier" title="Transformasi Fourier – Indonesian" lang="id" hreflang="id" data-title="Transformasi Fourier" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fourier%E2%80%93v%C3%B6rpun" title="Fourier–vörpun – Icelandic" lang="is" hreflang="is" data-title="Fourier–vörpun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier – Italian" lang="it" hreflang="it" data-title="Trasformata di Fourier" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9E%D7%A8%D7%AA_%D7%A4%D7%95%D7%A8%D7%99%D7%99%D7%94" title="התמרת פורייה – Hebrew" lang="he" hreflang="he" data-title="התמרת פורייה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/%D9%81%D9%88%D8%B1%DB%8C%D8%B1_%D9%B9%D8%B1%D8%A7%D9%86%D8%B3%D9%81%D8%A7%D8%B1%D9%85" title="فوریر ٹرانسفارم – Kashmiri" lang="ks" hreflang="ks" data-title="فوریر ٹرانسفارم" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="Kashmiri" class="interlanguage-link-target">कॉशुर / کٲشُر</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%82%D2%AF%D1%80%D0%BB%D0%B5%D0%BD%D0%B4%D1%96%D1%80%D1%83" title="Фурье түрлендіру – Kazakh" lang="kk" hreflang="kk" data-title="Фурье түрлендіру" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Furj%C4%97_transformacija" title="Furjė transformacija – Lithuanian" lang="lt" hreflang="lt" data-title="Furjė transformacija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fourier-transzform%C3%A1ci%C3%B3" title="Fourier-transzformáció – Hungarian" lang="hu" hreflang="hu" data-title="Fourier-transzformáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D0%B8%D0%B5%D0%BE%D0%B2%D0%B0_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B1%D0%B0" title="Фуриеова преобразба – Macedonian" lang="mk" hreflang="mk" data-title="Фуриеова преобразба" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Trasformata_ta%27_Fourier" title="Trasformata ta' Fourier – Maltese" lang="mt" hreflang="mt" data-title="Trasformata ta' Fourier" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%85%D1%83%D0%B2%D0%B8%D1%80%D0%B3%D0%B0%D0%BB%D1%82" title="Фурье хувиргалт – Mongolian" lang="mn" hreflang="mn" data-title="Фурье хувиргалт" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%96%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE%E1%80%9A%E1%80%AC_%E1%80%91%E1%80%9B%E1%80%94%E1%80%BA%E1%80%85%E1%80%96%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8" title="ဖိုရီယာ ထရန်စဖောင်း – Burmese" lang="my" hreflang="my" data-title="ဖိုရီယာ ထရန်စဖောင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Fouriertransformatie" title="Fouriertransformatie – Dutch" lang="nl" hreflang="nl" data-title="Fouriertransformatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%BC%E3%83%AA%E3%82%A8%E5%A4%89%E6%8F%9B" title="フーリエ変換 – Japanese" lang="ja" hreflang="ja" data-title="フーリエ変換" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fourier-transformasjon" title="Fourier-transformasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fourier-transformasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fourier-transformasjon" title="Fourier-transformasjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fourier-transformasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A9%8B%E0%A8%B0%E0%A9%80%E0%A8%85%E0%A8%B0_%E0%A8%AA%E0%A8%B0%E0%A8%BF%E0%A8%B5%E0%A8%B0%E0%A8%A4%E0%A8%A8" title="ਫੋਰੀਅਰ ਪਰਿਵਰਤਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਫੋਰੀਅਰ ਪਰਿਵਰਤਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Transformacja_Fouriera" title="Transformacja Fouriera – Polish" lang="pl" hreflang="pl" data-title="Transformacja Fouriera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transformada_de_Fourier" title="Transformada de Fourier – Portuguese" lang="pt" hreflang="pt" data-title="Transformada de Fourier" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transformata_Fourier" title="Transformata Fourier – Romanian" lang="ro" hreflang="ro" data-title="Transformata Fourier" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%A4%D1%83%D1%80%D1%8C%D0%B5" title="Преобразование Фурье – Russian" lang="ru" hreflang="ru" data-title="Преобразование Фурье" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Transformimi_i_Furierit" title="Transformimi i Furierit – Albanian" lang="sq" hreflang="sq" data-title="Transformimi i Furierit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fourier_transform" title="Fourier transform – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fourier transform" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fourierova_transform%C3%A1cia" title="Fourierova transformácia – Slovak" lang="sk" hreflang="sk" data-title="Fourierova transformácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fourierova_transformacija" title="Fourierova transformacija – Slovenian" lang="sl" hreflang="sl" data-title="Fourierova transformacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D0%B8%D1%98%D0%B5%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Фуријеова трансформација – Serbian" lang="sr" hreflang="sr" data-title="Фуријеова трансформација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical transform that expresses a function of time as a function of frequency</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with Fourier's original <a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">sine and cosine transforms</a>, which may be a simpler introduction to the Fourier transform.</div> <style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Fourier transforms</a></th></tr><tr><td class="sidebar-content"> <div class="plainlist"> <ul><li><a class="mw-selflink selflink">Fourier transform</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></li> <li><a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></li> <li><a href="/wiki/Discrete_Fourier_transform_(general)" class="mw-redirect" title="Discrete Fourier transform (general)">Discrete Fourier transform over a ring</a></li> <li><a href="/wiki/Fourier_transform_on_finite_groups" title="Fourier transform on finite groups">Fourier transform on finite groups</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Related transforms</a></li></ul> </div></td> </tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CQT-piano-chord.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/CQT-piano-chord.png/220px-CQT-piano-chord.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/CQT-piano-chord.png/330px-CQT-piano-chord.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/CQT-piano-chord.png/440px-CQT-piano-chord.png 2x" data-file-width="784" data-file-height="556" /></a><figcaption>An example application of the Fourier transform is determining the constituent pitches in a <a href="/wiki/Music" title="Music">musical</a> <a href="/wiki/Waveform" title="Waveform">waveform</a>. This image is the result of applying a <a href="/wiki/Constant-Q_transform" title="Constant-Q transform">constant-Q transform</a> (a <a href="/wiki/Fourier-related_transform" class="mw-redirect" title="Fourier-related transform">Fourier-related transform</a>) to the waveform of a <a href="/wiki/C_major" title="C major">C major</a> <a href="/wiki/Piano" title="Piano">piano</a> <a href="/wiki/Chord_(music)" title="Chord (music)">chord</a>. The first three peaks on the left correspond to the frequencies of the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> of the chord (C, E, G). The remaining smaller peaks are higher-frequency <a href="/wiki/Overtone" title="Overtone">overtones</a> of the fundamental pitches. A <a href="/wiki/Pitch_detection_algorithm" title="Pitch detection algorithm">pitch detection algorithm</a> could use the relative intensity of these peaks to infer which notes the pianist pressed.</figcaption></figure> In <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the Fourier transform (FT) is an <a href="/wiki/Integral_transform" title="Integral transform">integral transform</a> that takes a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a <a href="/wiki/Complex_number" title="Complex number">complex</a>-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">mathematical operation</a>. When a distinction needs to be made, the output of the operation is sometimes called the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> representation of the original function. The Fourier transform is analogous to decomposing the <a href="/wiki/Sound" title="Sound">sound</a> of a musical <a href="/wiki/Chord_(music)" title="Chord (music)">chord</a> into the <a href="/wiki/Sound_intensity" title="Sound intensity">intensities</a> of its constituent <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitches</a>. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Fourier_transform_time_and_frequency_domains_%28small%29.gif/220px-Fourier_transform_time_and_frequency_domains_%28small%29.gif" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/72/Fourier_transform_time_and_frequency_domains_%28small%29.gif 1.5x" data-file-width="300" data-file-height="240" /></a><figcaption>The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:116px;max-width:116px"><div class="thumbimage" style="height:118px;overflow:hidden"><a href="/wiki/File:Sine_voltage.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/114px-Sine_voltage.svg.png" decoding="async" width="114" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/171px-Sine_voltage.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/228px-Sine_voltage.svg.png 2x" data-file-width="816" data-file-height="850" /></a></div></div><div class="tsingle" style="width:172px;max-width:172px"><div class="thumbimage" style="height:118px;overflow:hidden"><a href="/wiki/File:Phase_shift.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Phase_shift.svg/170px-Phase_shift.svg.png" decoding="async" width="170" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Phase_shift.svg/255px-Phase_shift.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Phase_shift.svg/340px-Phase_shift.svg.png 2x" data-file-width="741" data-file-height="517" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The red <a href="/wiki/Sine_wave" title="Sine wave">sinusoid</a> can be described by peak amplitude (1), peak-to-peak (2), <a href="/wiki/Root_mean_square" title="Root mean square">RMS</a> (3), and <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> (4). The red and blue sinusoids have a phase difference of θ.</div></div></div></div>Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the <a href="#Uncertainty_principle">uncertainty principle</a>. The <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical</a> case for this principle is the <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a>, of substantial importance in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a> as well as in the study of physical phenomena exhibiting <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> (e.g., <a href="/wiki/Diffusion" title="Diffusion">diffusion</a>). The Fourier transform of a Gaussian function is another Gaussian function. <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> introduced <a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">sine and cosine transforms</a> (which <a href="/wiki/Sine_and_cosine_transforms#Relation_with_complex_exponentials" title="Sine and cosine transforms">correspond to the imaginary and real components</a> of the modern Fourier transform) in his study of <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, where Gaussian functions appear as solutions of the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>. The Fourier transform can be formally defined as an <a href="/wiki/Improper_integral" title="Improper integral">improper</a> <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a>, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.<a href="#cite_note-1">[note 1]</a> For example, many relatively simple applications use the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.<a href="#cite_note-2">[note 2]</a> The Fourier transform can also be generalized to functions of several variables on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of <a href="/wiki/4-momentum" class="mw-redirect" title="4-momentum">4-momentum</a>). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued</a>.<a href="#cite_note-3">[note 3]</a> Still further generalization is possible to functions on <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, which, besides the original Fourier transform on <a href="/wiki/Real_number#Arithmetic" title="Real number">R</a> or Rn, notably includes the <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT, group = <a href="/wiki/Integers" class="mw-redirect" title="Integers">Z</a>), the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT, group = <a href="/wiki/Cyclic_group" title="Cyclic group">Z mod N</a>) and the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> or circular Fourier transform (group = <a href="/wiki/Circle_group" title="Circle group">S1</a>, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a>. The <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) is an algorithm for computing the DFT. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> The Fourier transform is an analysis process, decomposing a complex-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fa998ae55408f7f94d08ee08a04fcf92330878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \textstyle f(x)}"> into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fa998ae55408f7f94d08ee08a04fcf92330878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \textstyle f(x)}"> from its transform. We can start with an analogy, the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, which analyzes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fa998ae55408f7f94d08ee08a04fcf92330878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle \textstyle f(x)}"> on a bounded interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x\in [-P/2,P/2],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle x\in [-P/2,P/2],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be4c2043b276e0f129991362775e5c5beff8982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.094ex; height:2.843ex;" alt="{\displaystyle \textstyle x\in [-P/2,P/2],}"> for some positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f4f085fcd14302f4f7a9bbdf77e816cccb3bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle P.}"> The constituent frequencies are a discrete set of harmonics at frequencies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{P}},n\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{P}},n\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5822d4d41e37acdd121bd07258c0b75b08b89a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.537ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{P}},n\in \mathbb {Z} ,}"> whose amplitude and phase are given by the analysis formula:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={\frac {1}{P}}\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>P</mi> </mfrac> </mrow> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {1}{P}}\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f8fcbd2cbbf828045ad99386b7d16f0e94aa9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.449ex; height:6.676ex;" alt="{\displaystyle c_{n}={\frac {1}{P}}\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.}">The actual Fourier series is the synthesis formula:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},\quad \textstyle x\in [-P/2,P/2].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},\quad \textstyle x\in [-P/2,P/2].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179ca2af800697798c172e1b58a650fbe2a1ccc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.81ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},\quad \textstyle x\in [-P/2,P/2].}">On an unbounded interval, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6cee49754c6d2a1bf8dd48b800bc438a5da57c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.33ex; height:2.509ex;" alt="{\displaystyle P\to \infty ,}"> the constituent frequencies are a continuum: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">→</mo> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de39ff3c77574e12261eccaebfa490c0ed745caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.88ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R} ,}"><a href="#cite_note-4">[1]</a><a href="#cite_note-5">[2]</a><a href="#cite_note-6">[3]</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"> is replaced by a function:<a href="#cite_note-7">[4]</a><div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">Fourier transform <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6878d38ad411d2000a6239e949737229cbdcfbc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.714ex; height:6.009ex;" alt="{\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx.}">     </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(Eq.1)</td></tr></tbody></table> </div> Evaluating <a href="#math_Eq.1">Eq.1</a> for all values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> produces the frequency-domain function. The integral can diverge at some frequencies. (see <a href="#Fourier_transform_for_periodic_functions">§ Fourier transform for periodic functions</a>) But it converges for all frequencies when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> decays with all derivatives as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \pm \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mo>±</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to \pm \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b105bb7c549379dd5714250e1b1afbdea39c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.076ex; height:2.176ex;" alt="{\displaystyle x\to \pm \infty }">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to \infty }f^{(n)}(x)=0,n=0,1,2,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f^{(n)}(x)=0,n=0,1,2,\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbbafc54b0f84ae6e43ddffe2e31e16bee271eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.671ex; height:4.343ex;" alt="{\displaystyle \lim _{x\to \infty }f^{(n)}(x)=0,n=0,1,2,\dots }">. (See <a href="/wiki/Schwartz_function" class="mw-redirect" title="Schwartz function">Schwartz function</a>). By the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>, the transformed function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> also decays with all derivatives. The complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf24289c9219fa2cfd300c1e50881fcb2089c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.61ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi )}">, in polar coordinates, conveys both <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> and <a href="/wiki/Phase_offset" class="mw-redirect" title="Phase offset">phase</a> of frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301954fda87cb533b5ff06a995680fb94c521266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.677ex; height:2.509ex;" alt="{\displaystyle \xi .}"> The intuitive interpretation of <a href="#math_Eq.1">Eq.1</a> is that the effect of multiplying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i2\pi \xi x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i2\pi \xi x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fb2df821c09da15777d0d410d97ba5186d63ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.676ex;" alt="{\displaystyle e^{-i2\pi \xi x}}"> is to subtract <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> from every frequency component of function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e889a7f3ee0216318cbf9f3478208fa1404cc51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x).}"><a href="#cite_note-8">[note 4]</a> Only the component that was at frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see <a href="#Example">§ Example</a>) The corresponding synthesis formula is: <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">Inverse transform <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b182002e69d633c3433092e8401ee2b0fcdd3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.214ex; height:6.009ex;" alt="{\displaystyle f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{i2\pi \xi x}\,d\xi ,\quad \forall \ x\in \mathbb {R} .}">     </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(Eq.2)</td></tr></tbody></table> </div> <a href="#math_Eq.2">Eq.2</a> is a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> as a weighted summation of complex exponential functions. This is also known as the <a href="/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a>, and was first introduced in <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier's</a> Analytical Theory of Heat.<a href="#cite_note-9">[5]</a><a href="#cite_note-10">[6]</a><a href="#cite_note-11">[7]</a><a href="#cite_note-12">[8]</a> The functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> are referred to as a Fourier transform pair.<a href="#cite_note-13">[9]</a>  A common notation for designating transform pairs is:<a href="#cite_note-14">[10]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟷</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b449cfeb06d29b6c104875b971fa00e6ce9c317" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.155ex; height:4.176ex;" alt="{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ),}">   for example   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟷</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e95bd2b0d7b3c2d31e1816383f25f5b65374b7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.261ex; height:4.176ex;" alt="{\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi ).}"> <div class="mw-heading mw-heading3"><h3 id="Definition_for_Lebesgue_integrable_functions">Definition for Lebesgue integrable functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=2" title="Edit section: Definition for Lebesgue integrable functions">edit</a>]</div> Until now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives. This excludes many functions of practical importance from the definition, such as the <a href="/wiki/Rect_function" class="mw-redirect" title="Rect function">rect function</a>. A <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fd4f6d5889adc68cfb7e6043cfc3cf8d0dd258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {C} }"> is called (Lebesgue) integrable if the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a> of its absolute value is finite: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594dfc8c7fcd0575cfa65ff750d563aa4875583a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.568ex; height:5.676ex;" alt="{\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty .}"> Two measurable functions are equivalent if they are equal except on a set of measure zero. The set of all equivalence classes of integrable functions is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a86223fba658a5daf612b39d3e25495d548bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} )}">. Then:<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> Definition — The Fourier transform of a Lebesgue integrable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{1}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{1}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b74a27234b1df8cb0ef6c17fd8062ac0ceef576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:3.176ex;" alt="{\displaystyle f\in L^{1}(\mathbb {R} )}"> is defined by the formula <a href="#math_Eq.1">Eq.1</a>. </div> The integral <a href="#math_Eq.1">Eq.1</a> is well-defined for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0edc17cdde255d4e4acb2243f4dd97b7c409b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.196ex; height:2.509ex;" alt="{\displaystyle \xi \in \mathbb {R} ,}"> because of the assumption <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{1}<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{1}<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90fdc01fc53547381048f3c9b6ccee484cd7c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.08ex; height:2.843ex;" alt="{\displaystyle \|f\|_{1}<\infty }">. (It can be shown that the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d9dc182926773ffe781af3061ff82283776b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.906ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R} )}"> is bounded and <a href="/wiki/Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a> in the frequency domain, and moreover, by the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>, it is zero at infinity.) However, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform. <div class="mw-heading mw-heading3"><h3 id="Unitarity_and_definition_for_square_integrable_functions">Unitarity and definition for square integrable functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=3" title="Edit section: Unitarity and definition for square integrable functions">edit</a>]</div> While <a href="#math_Eq.1">Eq.1</a> defines the Fourier transform for (complex-valued) functions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a86223fba658a5daf612b39d3e25495d548bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} )}">, it is easy to see that it is not well-defined for other integrability classes, most importantly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}">. For functions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}\cap L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>∩</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}\cap L^{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966fd2a56686cc8d2ff895cc58dd91932ec6e09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.344ex; height:3.176ex;" alt="{\displaystyle L^{1}\cap L^{2}(\mathbb {R} )}">, and with the conventions of <a href="#math_Eq.1">Eq.1</a>, the Fourier transform is a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a> with respect to the Hilbert inner product on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}">, restricted to the dense subspace of integrable functions. Therefore, it admits a unique continuous extension to a unitary operator on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}">, also called the Fourier transform. This extension is important in part because the Fourier transform preserves the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}"> so that, unlike the case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}">, the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself. Importantly, for functions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}">, the Fourier transform is no longer given by <a href="#math_Eq.1">Eq.1</a> (interpreted as a Lebesgue integral). For example, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(1+x^{2})^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(1+x^{2})^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef0888ad24791919f4758109326efb4e5bcaf80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.689ex; height:3.343ex;" alt="{\displaystyle f(x)=(1+x^{2})^{-1/2}}"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> but not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}">, so the integral <a href="#math_Eq.1">Eq.1</a> diverges. In such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit. In practice, the integral is often regarded as an <a href="/wiki/Improper_integral" title="Improper integral">improper integral</a> instead of a proper Lebesgue integral, but sometimes for convergence one needs to use <a href="/wiki/Weak_limit" class="mw-redirect" title="Weak limit">weak limit</a> or <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">principal value</a> instead of the (pointwise) limits implicit in an improper integral. <a href="#CITEREFTitchmarsh1986">Titchmarsh (1986)</a> and <a href="#CITEREFDymMcKean1985">Dym & McKean (1985)</a> each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure. The conventions chosen in this article are those of <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>, and are characterized as the unique conventions such that the Fourier transform is both <a href="/wiki/Unitary_operator" title="Unitary operator">unitary</a> on L2 and an algebra homomorphism from L1 to L∞, without renormalizing the Lebesgue measure.<a href="#cite_note-16">[12]</a> <div class="mw-heading mw-heading3"><h3 id="Angular_frequency_(ω)">Angular frequency (ω)</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=4" title="Edit section: Angular frequency (ω)">edit</a>]</div> When the independent variable (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">) represents time (often denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">), the transform variable (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }">) represents <a href="/wiki/Frequency" title="Frequency">frequency</a> (often denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">). For example, if time is measured in <a href="/wiki/Second" title="Second">seconds</a>, then frequency is in <a href="/wiki/Hertz" title="Hertz">hertz</a>. The Fourier transform can also be written in terms of <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi \xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi \xi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772e59ac44e71e27c1b262d85ea4e2b945fbeae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.716ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi \xi ,}"> whose units are <a href="/wiki/Radian" title="Radian">radians</a> per second. The substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ={\tfrac {\omega }{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ={\tfrac {\omega }{2\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df73ed29b9da8b21d7153501885e6ee3a80ef8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.729ex; height:3.176ex;" alt="{\displaystyle \xi ={\tfrac {\omega }{2\pi }}}"> into <a href="#math_Eq.1">Eq.1</a> produces this convention, where function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> is relabeled <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f_{1}}}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f_{1}}}:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a027cd86497dad0923847dbcd144da232f50054d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:3.626ex; height:3.343ex;" alt="{\displaystyle {\widehat {f_{1}}}:}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )&\triangleq \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )&\triangleq \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ce33fd475ca20de0a57c101996bfa2ae17601f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:40.635ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )&\triangleq \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}"> Unlike the <a href="#math_Eq.1">Eq.1</a> definition, the Fourier transform is no longer a <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformation</a>, and there is less symmetry between the formulas for the transform and its inverse. Those properties are restored by splitting the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> factor evenly between the transform and its inverse, which leads to another convention: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )&\triangleq {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )&\triangleq {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9978c58af0279004978ad1d8f01dc30f6d82bd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:52.716ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )&\triangleq {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\left({\tfrac {\omega }{2\pi }}\right),\\f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\cdot e^{i\omega x}\,d\omega .\end{aligned}}}"> Variations of all three conventions can be created by conjugating the complex-exponential <a href="/wiki/Integral_kernel" class="mw-redirect" title="Integral kernel">kernel</a> of both the forward and the reverse transform. The signs must be opposites. <table class="wikitable"> <caption>Summary of popular forms of the Fourier transform, one-dimensional </caption> <tbody><tr> <th>ordinary frequency ξ (Hz) </th> <th>unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i2\pi \xi x}\,dx={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{-\infty }^{\infty }{\widehat {f_{1}}}(\xi )\,e^{i2\pi x\xi }\,d\xi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i2\pi \xi x}\,dx={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{-\infty }^{\infty }{\widehat {f_{1}}}(\xi )\,e^{i2\pi x\xi }\,d\xi \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b92ac8ef312fe6215da92fc4bcaead4683714f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:56.83ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i2\pi \xi x}\,dx={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{-\infty }^{\infty }{\widehat {f_{1}}}(\xi )\,e^{i2\pi x\xi }\,d\xi \end{aligned}}}"> </td></tr> <tr> <th rowspan="2">angular frequency ω (rad/s) </th> <th>unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74fad886f3cc1bf7eb573180ec41d2e793319758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:67.581ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{\sqrt {2\pi }}}\ \ {\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{\sqrt {2\pi }}}\ \int _{-\infty }^{\infty }{\widehat {f_{2}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}"> </td></tr> <tr> <th>non-unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc405b5876944ab4cc47aaf35ab0fbd3decefb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:54.857ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{-\infty }^{\infty }f(x)\,e^{-i\omega x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)={\sqrt {2\pi }}\ \ {\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f_{3}}}(\omega )\,e^{i\omega x}\,d\omega \end{aligned}}}"> </td></tr></tbody></table> <table class="wikitable"> <caption>Generalization for n-dimensional functions </caption> <tbody><tr> <th>ordinary frequency ξ (Hz) </th> <th>unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{\mathbb {R} ^{n}}{\widehat {f_{1}}}(\xi )e^{i2\pi \xi \cdot x}\,d\xi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{\mathbb {R} ^{n}}{\widehat {f_{1}}}(\xi )e^{i2\pi \xi \cdot x}\,d\xi \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5e7867bf91faf442de73728c12482cd7a5b722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:56.711ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{1}}}(\xi )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(2\pi \xi )={\widehat {f_{3}}}(2\pi \xi )\\f(x)&=\int _{\mathbb {R} ^{n}}{\widehat {f_{1}}}(\xi )e^{i2\pi \xi \cdot x}\,d\xi \end{aligned}}}"> </td></tr> <tr> <th rowspan="2">angular frequency ω (rad/s) </th> <th>unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{2}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{2}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392222b17168cad2b530cc33dcc7975640cc1357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:69.205ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{2}}}(\omega )\ &\triangleq \ {\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{1}}}\!\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{\frac {n}{2}}}}{\widehat {f_{3}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{2}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}"> </td></tr> <tr> <th>non-unitary </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{3}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mtext> </mtext> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ω</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{3}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156fae48fb4c5ef72f380385b7c8d04b041278b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:54.738ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f_{3}}}(\omega )\ &\triangleq \ \int _{\mathbb {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx={\widehat {f_{1}}}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{\frac {n}{2}}{\widehat {f_{2}}}(\omega )\\f(x)&={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\widehat {f_{3}}}(\omega )e^{i\omega \cdot x}\,d\omega \end{aligned}}}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Extension_of_the_definition">Extension of the definition</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=5" title="Edit section: Extension of the definition">edit</a>]</div> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<p<2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8a390e08862f3f4a52a2855051202494ef8752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 1<p<2}">, the Fourier transform can be defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b25cc9016efea65c3a2be0b1a358b0d399ce3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} )}"> by <a href="/wiki/Marcinkiewicz_interpolation" class="mw-redirect" title="Marcinkiewicz interpolation">Marcinkiewicz interpolation</a>. The Fourier transform can be defined on domains other than the real line. The <a href="#Fourier_transform_on_Euclidean_space">Fourier transform on Euclidean space</a> and the <a href="#Locally_compact_abelian_groups">Fourier transform on locally abelian groups</a> are discussed later in the article. The Fourier transform can also be defined for <a href="/wiki/Tempered_distribution" class="mw-redirect" title="Tempered distribution">tempered distributions</a>, dual to the space of rapidly decreasing functions (<a href="/wiki/Schwartz_function" class="mw-redirect" title="Schwartz function">Schwartz functions</a>). A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives. The space of Schwartz functions is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0751119ebd490eacaceee66c787196ffae55c316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.98ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} )}">, and its dual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8c5377581a431b919b6170e94f771d69583a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.69ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} )}"> is the space of tempered distributions. It is easy to see, by differentiating under the integral and applying the Riemann-Lebesgue lemma, that the Fourier transform of a Schwartz function (defined by the formula <a href="#math_Eq.1">Eq.1</a>) is again a Schwartz function. The Fourier transform of a tempered distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd66f3b1b935ab2b29cfbcd8e1b105a220a4deca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.166ex; height:3.009ex;" alt="{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} )}"> is defined by duality: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\widehat {T}},\phi \rangle =\langle T,{\widehat {\phi }}\rangle ;\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>;</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>ϕ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\widehat {T}},\phi \rangle =\langle T,{\widehat {\phi }}\rangle ;\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224649b5f5d437c22d447c0bc4eb2c8bdec8a759" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.68ex; height:3.343ex;" alt="{\displaystyle \langle {\widehat {T}},\phi \rangle =\langle T,{\widehat {\phi }}\rangle ;\quad \forall \phi \in {\mathcal {S}}(\mathbb {R} ).}"> Many other characterizations of the Fourier transform exist. For example, one uses the <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>: the Fourier transform is the unique unitary <a href="/wiki/Intertwiner" class="mw-redirect" title="Intertwiner">intertwiner</a> for the symplectic and Euclidean Schrödinger representations of the <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a>. <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=6" title="Edit section: Background">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="History">History</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=7" title="Edit section: History">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Fourier_analysis#History" title="Fourier analysis">Fourier analysis § History</a>, and <a href="/wiki/Fourier_series#History" title="Fourier series">Fourier series § History</a></div> In 1822, Fourier claimed (see <a href="/wiki/Joseph_Fourier#The_Analytic_Theory_of_Heat" title="Joseph Fourier">Joseph Fourier § The Analytic Theory of Heat</a>) that any function, whether continuous or discontinuous, can be expanded into a series of sines.<a href="#cite_note-17">[13]</a> That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Unfasor.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/89/Unfasor.gif" decoding="async" width="219" height="432" class="mw-file-element" data-file-width="219" data-file-height="432" /></a><figcaption>Fig.1 When function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot e^{i2\pi \xi t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot e^{i2\pi \xi t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7af52cdc5f71d361cb03a969826200a5ce6d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.391ex; height:2.676ex;" alt="{\displaystyle A\cdot e^{i2\pi \xi t}}"> is depicted in the complex plane, the vector formed by its <a href="/wiki/Complex_number" title="Complex number">imaginary and real parts</a> rotates around the origin. Its real part <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}"> is a cosine wave.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Complex_sinusoids">Complex sinusoids</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=8" title="Edit section: Complex sinusoids">edit</a>]</div> In general, the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf24289c9219fa2cfd300c1e50881fcb2089c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.61ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi )}"> are complex numbers, which have two equivalent forms (see <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>polar coordinate form</mtext> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rectangular coordinate form</mtext> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1afd750cfb93bfa43d1a25cc0a47c174ca0afdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:47.246ex; height:7.343ex;" alt="{\displaystyle {\widehat {f}}(\xi )=\underbrace {Ae^{i\theta }} _{\text{polar coordinate form}}=\underbrace {A\cos(\theta )+iA\sin(\theta )} _{\text{rectangular coordinate form}}.}"> The product with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi \xi x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi \xi x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21b0e3490934b365ffe026bd724c2a029a3bc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle e^{i2\pi \xi x}}"> (<a href="#math_Eq.2">Eq.2</a>) has these forms: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {f}}(\xi )\cdot e^{i2\pi \xi x}&=Ae^{i\theta }\cdot e^{i2\pi \xi x}\\&=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}\\&=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>polar coordinate form</mtext> </mrow> </munder> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rectangular coordinate form</mtext> </mrow> </munder> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {f}}(\xi )\cdot e^{i2\pi \xi x}&=Ae^{i\theta }\cdot e^{i2\pi \xi x}\\&=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}\\&=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d4431fd50d8cf141cec3c2369a5dbd70c052e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:51.161ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {f}}(\xi )\cdot e^{i2\pi \xi x}&=Ae^{i\theta }\cdot e^{i2\pi \xi x}\\&=\underbrace {Ae^{i(2\pi \xi x+\theta )}} _{\text{polar coordinate form}}\\&=\underbrace {A\cos(2\pi \xi x+\theta )+iA\sin(2\pi \xi x+\theta )} _{\text{rectangular coordinate form}}.\end{aligned}}}"> It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula. <div class="mw-heading mw-heading3"><h3 id="Negative_frequency">Negative frequency</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=9" title="Edit section: Negative frequency">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Negative_frequency#Simplifying_the_Fourier_transform" title="Negative frequency">Negative frequency § Simplifying the Fourier transform</a></div> Euler's formula introduces the possibility of negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301954fda87cb533b5ff06a995680fb94c521266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.677ex; height:2.509ex;" alt="{\displaystyle \xi .}">  And <a href="#math_Eq.1">Eq.1</a> is defined <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \xi \in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \xi \in \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de562c799ca7f9ee87746c1747069d0d3fef81d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.488ex; height:2.509ex;" alt="{\displaystyle \forall \xi \in \mathbb {R} .}"> Only certain complex-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> have transforms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}=0,\ \forall \ \xi <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>ξ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}=0,\ \forall \ \xi <0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db05e7dd64ab70f26c4ae8212aaa1bd641c8e1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.811ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}=0,\ \forall \ \xi <0}"> (See <a href="/wiki/Analytic_signal" title="Analytic signal">Analytic signal</a>. A simple example is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi \xi _{0}x}\ (\xi _{0}>0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi \xi _{0}x}\ (\xi _{0}>0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a34a21c666866f8354d967ce7744c5e88faaa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.509ex; height:3.176ex;" alt="{\displaystyle e^{i2\pi \xi _{0}x}\ (\xi _{0}>0).}">)  But negative frequency is necessary to characterize all other complex-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"> found in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, <a href="/wiki/Radar" title="Radar">radar</a>, <a href="/wiki/Nonlinear_optics" title="Nonlinear optics">nonlinear optics</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, and others. For a real-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"> <a href="#math_Eq.1">Eq.1</a> has the symmetry property <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\xi )={\widehat {f}}^{*}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\xi )={\widehat {f}}^{*}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50a670cb111a7210bfb3e369d2bd4416c5fa50da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.182ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(-\xi )={\widehat {f}}^{*}(\xi )}"> (see <a href="#Conjugation">§ Conjugation</a> below). This redundancy enables <a href="#math_Eq.2">Eq.2</a> to distinguish <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\cos(2\pi \xi _{0}x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\cos(2\pi \xi _{0}x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9469b2564627e7f2565c32971b9810e5da5d9e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.333ex; height:2.843ex;" alt="{\displaystyle f(x)=\cos(2\pi \xi _{0}x)}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi \xi _{0}x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi \xi _{0}x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777ea4d98a632a4c4ae9a0aeebb254657bf5b84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.786ex; height:2.676ex;" alt="{\displaystyle e^{i2\pi \xi _{0}x}.}">  But of course it cannot tell us the actual sign of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25bd4fe87ec09e09111556de6d212964c82bfa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.509ex;" alt="{\displaystyle \xi _{0},}"> because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi \xi _{0}x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi \xi _{0}x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf7c101643f84d7c3ede82a512b33fea021561d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.817ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi \xi _{0}x)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi (-\xi _{0})x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi (-\xi _{0})x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b5d11bc741efd26be0eaa4361937efc7bb5fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.435ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi (-\xi _{0})x)}"> are indistinguishable on just the real numbers line. <div class="mw-heading mw-heading3"><h3 id="Fourier_transform_for_periodic_functions">Fourier transform for periodic functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=10" title="Edit section: Fourier transform for periodic functions">edit</a>]</div> The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in <a href="#math_Eq.1">Eq.1</a> to be defined the function must be <a href="/wiki/Absolutely_integrable_function" title="Absolutely integrable function">absolutely integrable</a>. Instead it is common to use <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. It is possible to extend the definition to include periodic functions by viewing them as <a href="/wiki/Distribution_(mathematics)#Tempered_distributions" title="Distribution (mathematics)">tempered distributions</a>. This makes it possible to see a connection between the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> and the Fourier transform for periodic functions that have a <a href="/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">convergent Fourier series</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is a <a href="/wiki/Periodic_function" title="Periodic function">periodic function</a>, with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">, that has a convergent Fourier series, then: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ceef16aa3bef9aba66f10a0b188a55a905992d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.373ex; height:6.843ex;" alt="{\displaystyle {\widehat {f}}(\xi )=\sum _{n=-\infty }^{\infty }c_{n}\cdot \delta \left(\xi -{\tfrac {n}{P}}\right),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"> are the Fourier series coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. In other words, the Fourier transform is a <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a> function whose teeth are multiplied by the Fourier series coefficients. <div class="mw-heading mw-heading3"><h3 id="Sampling_the_Fourier_transform">Sampling the Fourier transform</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=11" title="Edit section: Sampling the Fourier transform">edit</a>]</div> The Fourier transform of an <a href="/wiki/Absolutely_integrable_function" title="Absolutely integrable function">integrable</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> can be sampled at regular intervals of arbitrary length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{P}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{P}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe880fb45edd7850339811b311fd3f83bdd08a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.717ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{P}}.}"> These samples can be deduced from one cycle of a periodic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43bdbe4ab8c7bbbb89a5410c25b536d10befbb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.606ex; height:2.509ex;" alt="{\displaystyle f_{P}}"> which has <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> coefficients proportional to those samples by the <a href="/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\widehat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>x</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\widehat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda40763487a38c721fcd3ca5733bb378dc23b51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.393ex; height:7.009ex;" alt="{\displaystyle f_{P}(x)\triangleq \sum _{n=-\infty }^{\infty }f(x+nP)={\frac {1}{P}}\sum _{k=-\infty }^{\infty }{\widehat {f}}\left({\tfrac {k}{P}}\right)e^{i2\pi {\frac {k}{P}}x},\quad \forall k\in \mathbb {Z} }"> The integrability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> ensures the periodic summation converges. Therefore, the samples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f36468faab271d3d9f9bd6371c421f86d52c248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.004ex; height:4.843ex;" alt="{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)}"> can be determined by Fourier series analysis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c22c792da81986c2443240c1c9ab8d7ced254d4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.956ex; height:5.676ex;" alt="{\displaystyle {\widehat {f}}\left({\tfrac {k}{P}}\right)=\int _{P}f_{P}(x)\cdot e^{-i2\pi {\frac {k}{P}}x}\,dx.}"> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> has <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b4d293b8559025b05cdeabd847ece92e33e8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.745ex; height:2.843ex;" alt="{\displaystyle f_{P}(x)}"> has a finite number of terms within the interval of integration. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> does not have compact support, numerical evaluation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b4d293b8559025b05cdeabd847ece92e33e8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.745ex; height:2.843ex;" alt="{\displaystyle f_{P}(x)}"> requires an approximation, such as tapering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> or truncating the number of terms. <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=12" title="Edit section: Example">edit</a>]</div> The following figures provide a visual illustration of how the Fourier transform's integral measures whether a frequency is present in a particular function. The first image depicts the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mn>3</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23a2291ba2775682f5f89bb818a880af3d456e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.212ex; height:3.509ex;" alt="{\displaystyle f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}},}"> which is a 3 <a href="/wiki/Hertz" title="Hertz">Hz</a> cosine wave (the first term) shaped by a <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian</a> <a href="/wiki/Envelope_(waves)" title="Envelope (waves)">envelope function</a> (the second term) that smoothly turns the wave on and off. The next 2 images show the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)e^{-i2\pi 3t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)e^{-i2\pi 3t},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72c8055f8e22a17ed0e188522a056e29d1d3d3bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.916ex; height:3.176ex;" alt="{\displaystyle f(t)e^{-i2\pi 3t},}"> which must be integrated to calculate the Fourier transform at +3 Hz. The real part of the integrand has a non-negative average value, because the alternating signs of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (e^{-i2\pi 3t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (e^{-i2\pi 3t})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7c5dfe8df5d7dd2caf85bafab65f7bf15b4570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.894ex; height:3.176ex;" alt="{\displaystyle \operatorname {Re} (e^{-i2\pi 3t})}"> oscillate at the same rate and in phase, whereas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (e^{-i2\pi 3t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (e^{-i2\pi 3t})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b7a3fd61b28e9ab6fdc7e9eb4a18617789df74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.926ex; height:3.176ex;" alt="{\displaystyle \operatorname {Im} (e^{-i2\pi 3t})}"> oscillate at the same rate but with orthogonal phase. The absolute value of the Fourier transform at +3 Hz is 0.5, which is relatively large. When added to the Fourier transform at -3 Hz (which is identical because we started with a real signal), we find that the amplitude of the 3 Hz frequency component is 1. <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Onfreq.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Onfreq.png/695px-Onfreq.png" decoding="async" width="695" height="324" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Onfreq.png/1043px-Onfreq.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2f/Onfreq.png 2x" data-file-width="1274" data-file-height="594" /></a><figcaption>Original function, which has a strong 3 Hz component. Real and imaginary parts of the integrand of its Fourier transform at +3 Hz.</figcaption></figure> <div style="clear:both;" class=""></div>However, when you try to measure a frequency that is not present, both the real and imaginary component of the integral vary rapidly between positive and negative values. For instance, the red curve is looking for 5 Hz. The absolute value of its integral is nearly zero, indicating that almost no 5 Hz component was in the signal. The general situation is usually more complicated than this, but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db88c28d6c644c905a4e12de7971c3d1eed37540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.574ex; height:2.843ex;" alt="{\displaystyle f(t).}"><ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 395px"> <div class="thumb" style="width: 390px; height: 390px;"><a href="/wiki/File:Offfreq_i2p.svg" class="mw-file-description" title="Real and imaginary parts of the integrand for its Fourier transform at +5 Hz."><img alt="Real and imaginary parts of the integrand for its Fourier transform at +5 Hz." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Offfreq_i2p.svg/360px-Offfreq_i2p.svg.png" decoding="async" width="360" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Offfreq_i2p.svg/540px-Offfreq_i2p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Offfreq_i2p.svg/720px-Offfreq_i2p.svg.png 2x" data-file-width="575" data-file-height="495" /></a></div> <div class="gallerytext"> Real and imaginary parts of the integrand for its Fourier transform at +5 Hz.</div> </li> <li class="gallerybox" style="width: 395px"> <div class="thumb" style="width: 390px; height: 390px;"><a href="/wiki/File:Fourier_transform_of_oscillating_function.svg" class="mw-file-description" title="Magnitude of its Fourier transform, with +3 and +5 Hz labeled."><img alt="Magnitude of its Fourier transform, with +3 and +5 Hz labeled." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fourier_transform_of_oscillating_function.svg/360px-Fourier_transform_of_oscillating_function.svg.png" decoding="async" width="360" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fourier_transform_of_oscillating_function.svg/540px-Fourier_transform_of_oscillating_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fourier_transform_of_oscillating_function.svg/720px-Fourier_transform_of_oscillating_function.svg.png 2x" data-file-width="598" data-file-height="494" /></a></div> <div class="gallerytext"> Magnitude of its Fourier transform, with +3 and +5 Hz labeled.</div> </li> </ul> To re-enforce an earlier point, the reason for the response at  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =-3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fc7cce49f684f02fd9f7c3a864502a21fbd72d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.099ex; height:2.509ex;" alt="{\displaystyle \xi =-3}"> Hz  is because  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi 3t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi 3t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5cf99465284f29aff98d05b97d1320a29fb13ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.417ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi 3t)}">  and  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi (-3)t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi (-3)t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07336ab6aef6c1ca5243bd274fa688c71f76aa3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.034ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi (-3)t)}">  are indistinguishable. The transform of  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi 3t}\cdot e^{-\pi t^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi 3t}\cdot e^{-\pi t^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ad7480778f419dac238a3a83a1bdcebf6bbdc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.703ex; height:3.009ex;" alt="{\displaystyle e^{i2\pi 3t}\cdot e^{-\pi t^{2}}}">  would have just one response, whose amplitude is the integral of the smooth envelope: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi t^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi t^{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a3a326b85fc65d085122006384f111eae35695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.608ex; height:3.343ex;" alt="{\displaystyle e^{-\pi t^{2}},}">  whereas  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56bc1a5f371a70dc3aa714809df24f4b4a9f9f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.5ex; height:3.176ex;" alt="{\displaystyle \operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})}"> is  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mn>6</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84099d8e713424452ec9e5c55827d168406ea0d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.162ex; height:3.509ex;" alt="{\displaystyle e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.}"> <div class="mw-heading mw-heading2"><h2 id="Properties_of_the_Fourier_transform">Properties of the Fourier transform</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=13" title="Edit section: Properties of the Fourier transform">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c07825dae28705df03d15daeb8844d49c4dbd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.478ex; height:2.843ex;" alt="{\displaystyle h(x)}"> represent integrable functions <a href="/wiki/Lebesgue-measurable" class="mw-redirect" title="Lebesgue-measurable">Lebesgue-measurable</a> on the real line satisfying: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18053041a2f48527e792f7f6f42a30c486d4fd6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.547ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|f(x)|\,dx<\infty .}"> We denote the Fourier transforms of these functions as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c12179a82497132d68a556a537ef57b46cbbf83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.538ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10388d495d1f3bddd53259ce4ce6a82ec81ce908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.178ex; height:3.343ex;" alt="{\displaystyle {\hat {h}}(\xi )}"> respectively. <div class="mw-heading mw-heading3"><h3 id="Basic_properties">Basic properties</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=14" title="Edit section: Basic properties">edit</a>]</div> The Fourier transform has the following basic properties:<a href="#cite_note-Pinsky-2002-18">[14]</a> <div class="mw-heading mw-heading4"><h4 id="Linearity">Linearity</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=15" title="Edit section: Linearity">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ f(x)+b\ h(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\ {\widehat {f}}(\xi )+b\ {\widehat {h}}(\xi );\quad \ a,b\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mtext> </mtext> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mtext> </mtext> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ f(x)+b\ h(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\ {\widehat {f}}(\xi )+b\ {\widehat {h}}(\xi );\quad \ a,b\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e32218dabd895e7e588d5b0240388ff026d78382" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.499ex; height:4.176ex;" alt="{\displaystyle a\ f(x)+b\ h(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ a\ {\widehat {f}}(\xi )+b\ {\widehat {h}}(\xi );\quad \ a,b\in \mathbb {C} }"> <div class="mw-heading mw-heading4"><h4 id="Time_shifting">Time shifting</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=16" title="Edit section: Time shifting">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x-x_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi x_{0}\xi }\ {\widehat {f}}(\xi );\quad \ x_{0}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ξ</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x-x_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi x_{0}\xi }\ {\widehat {f}}(\xi );\quad \ x_{0}\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3237cdaf3afec99d5809287835baf287a1fbcfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.738ex; height:4.176ex;" alt="{\displaystyle f(x-x_{0})\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ e^{-i2\pi x_{0}\xi }\ {\widehat {f}}(\xi );\quad \ x_{0}\in \mathbb {R} }"> <div class="mw-heading mw-heading4"><h4 id="Frequency_shifting">Frequency shifting</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=17" title="Edit section: Frequency shifting">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi \xi _{0}x}f(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(\xi -\xi _{0});\quad \ \xi _{0}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="1em" /> <mtext> </mtext> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi \xi _{0}x}f(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(\xi -\xi _{0});\quad \ \xi _{0}\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1cd0d60e96787f948603907e6b39e06f6af59d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.248ex; height:4.176ex;" alt="{\displaystyle e^{i2\pi \xi _{0}x}f(x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(\xi -\xi _{0});\quad \ \xi _{0}\in \mathbb {R} }"> <div class="mw-heading mw-heading4"><h4 id="Time_scaling">Time scaling</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=18" title="Edit section: Time scaling">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(ax)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right);\quad \ a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>;</mo> <mspace width="1em" /> <mtext> </mtext> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(ax)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right);\quad \ a\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53962c42205e7f89988998892dcd4fb6d6a9f49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.106ex; height:6.343ex;" alt="{\displaystyle f(ax)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right);\quad \ a\neq 0}"> The case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231103d8099e125875dd690668e93a56aa10bd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.299ex; height:2.343ex;" alt="{\displaystyle a=-1}"> leads to the time-reversal property: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(-\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(-\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/634cd0f2b232c72f0f76cec719d57a401336018a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.283ex; height:4.176ex;" alt="{\displaystyle f(-x)\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\widehat {f}}(-\xi )}"> <div class="noresize thumb tright" style=";"> <div class="thumbinner" style="overflow:hidden;width:302px;"> <div class="thumbimage" style="overflow:hidden; position:relative; background-color:white;"> <div style=";left:0px; top:0px; width:300px; position:absolute;"> <a href="/wiki/File:Fourier_unit_pulse.svg" class="mw-file-description" title="The transform of an even-symmetric real-valued function '"`UNIQ--postMath-00000086-QINU`"' is also an even-symmetric real-valued function '"`UNIQ--postMath-00000087-QINU`"' The time-shift, '"`UNIQ--postMath-00000088-QINU`"' creates an imaginary component, '"`UNIQ--postMath-00000089-QINU`"' (see § Symmmetry."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/300px-Fourier_unit_pulse.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/450px-Fourier_unit_pulse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/600px-Fourier_unit_pulse.svg.png 2x" data-file-width="800" data-file-height="600" /></a></div> <div style="text-align:left; background-color:transparent; line-height:110%;"> <div id="annotation_20x40" style="position:absolute; left:20px; top:40px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7646cfe88784c1c5cbd6d8432573fe074436b562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.777ex; height:2.176ex;" alt="{\displaystyle \scriptstyle f(t)}"></div> <div id="annotation_170x40" style="position:absolute; left:170px; top:40px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\widehat {f}}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\widehat {f}}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6191f91db7f79bdff527b11bc39d829bb3485c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.668ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\widehat {f}}(\omega )}"></div> <div id="annotation_20x140" style="position:absolute; left:20px; top:140px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/692893150c829aadf1f000c536fb815434c61963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.662ex; height:2.176ex;" alt="{\displaystyle \scriptstyle g(t)}"></div> <div id="annotation_170x140" style="position:absolute; left:170px; top:140px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\widehat {g}}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\widehat {g}}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b42fd34a11cb78603b29d39b4f7b0cac44cb69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.192ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {\widehat {g}}(\omega )}"></div> <div id="annotation_130x80" style="position:absolute; left:130px; top:80px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb2910bbfdfd61ced9d9519beb8f978af340d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.594ex; height:1.676ex;" alt="{\displaystyle \scriptstyle t}"></div> <div id="annotation_280x85" style="position:absolute; left:280px; top:85px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>ω</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691e776df98eed20b0084e0cb2953e837b45dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.022ex; height:1.343ex;" alt="{\displaystyle \scriptstyle \omega }"></div> <div id="annotation_130x192" style="position:absolute; left:130px; top:192px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb2910bbfdfd61ced9d9519beb8f978af340d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.594ex; height:1.676ex;" alt="{\displaystyle \scriptstyle t}"></div> <div id="annotation_280x180" style="position:absolute; left:280px; top:180px; line-height:110%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>ω</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691e776df98eed20b0084e0cb2953e837b45dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.022ex; height:1.343ex;" alt="{\displaystyle \scriptstyle \omega }"></div> </div> <div style="visibility:hidden"><a href="/wiki/File:Fourier_unit_pulse.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/300px-Fourier_unit_pulse.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/450px-Fourier_unit_pulse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Fourier_unit_pulse.svg/600px-Fourier_unit_pulse.svg.png 2x" data-file-width="800" data-file-height="600" /></a></div> </div> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Fourier_unit_pulse.svg" title="File:Fourier unit pulse.svg"> </a></div>The transform of an even-symmetric real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(t)=f_{RE})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f(t)=f_{RE})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d83b464fa6b9e28b3b9ab88afa94e289a4a1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.71ex; height:2.843ex;" alt="{\displaystyle (f(t)=f_{RE})}"> is also an even-symmetric real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {f}}_{RE}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {f}}_{RE}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d407313e09e0ac40a41f85ab6f79a92751667b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.89ex; height:3.343ex;" alt="{\displaystyle ({\hat {f}}_{RE}).}"> The time-shift, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g(t)=g_{RE}+g_{RO}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>O</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g(t)=g_{RE}+g_{RO}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581fa244b7c3ab480c53d3e564e55a6b6228bb39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.847ex; height:2.843ex;" alt="{\displaystyle (g(t)=g_{RE}+g_{RO}),}"> creates an imaginary component, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\cdot {\hat {g}}_{IO}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>O</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\cdot {\hat {g}}_{IO}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8a27a4e99dce07953b2430373b16a9dbc57591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.675ex; height:2.676ex;" alt="{\displaystyle i\cdot {\hat {g}}_{IO}.}"> (see <a href="#Symmmetry">§ Symmmetry</a>.</div> </div></div> <div class="mw-heading mw-heading4"><h4 id="Symmetry">Symmetry</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=19" title="Edit section: Symmetry">edit</a>]</div> When the real and imaginary parts of a complex function are decomposed into their <a href="/wiki/Even_and_odd_functions#Even–odd_decomposition" title="Even and odd functions">even and odd parts</a>, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rlcccccccc}{\mathsf {Time\ domain}}&f&=&f_{_{\text{RE}}}&+&f_{_{\text{RO}}}&+&i\ f_{_{\text{IE}}}&+&\underbrace {i\ f_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}&{\widehat {f}}&=&{\widehat {f}}_{_{\text{RE}}}&+&\overbrace {i\ {\widehat {f}}_{_{\text{IO}}}\,} &+&i\ {\widehat {f}}_{_{\text{IE}}}&+&{\widehat {f}}_{_{\text{RO}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left center center center center center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">e</mi> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">d</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </mtd> <mtd> <mi>f</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RE</mtext> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RO</mtext> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>IE</mtext> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>IO</mtext> </mrow> </msub> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">⇕</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">⇕</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">⇕</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">⇕</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">⇕</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">F</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">q</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">y</mi> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">d</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">m</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RE</mtext> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>i</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>IO</mtext> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> <mo>⏞</mo> </mover> </mrow> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mi>i</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>IE</mtext> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RO</mtext> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rlcccccccc}{\mathsf {Time\ domain}}&f&=&f_{_{\text{RE}}}&+&f_{_{\text{RO}}}&+&i\ f_{_{\text{IE}}}&+&\underbrace {i\ f_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}&{\widehat {f}}&=&{\widehat {f}}_{_{\text{RE}}}&+&\overbrace {i\ {\widehat {f}}_{_{\text{IO}}}\,} &+&i\ {\widehat {f}}_{_{\text{IE}}}&+&{\widehat {f}}_{_{\text{RO}}}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8c89b9ef8896bcf2044ab86985ebc7566c0bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.786ex; margin-bottom: -0.219ex; width:69.535ex; height:17.176ex;" alt="{\displaystyle {\begin{array}{rlcccccccc}{\mathsf {Time\ domain}}&f&=&f_{_{\text{RE}}}&+&f_{_{\text{RO}}}&+&i\ f_{_{\text{IE}}}&+&\underbrace {i\ f_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}&{\widehat {f}}&=&{\widehat {f}}_{_{\text{RE}}}&+&\overbrace {i\ {\widehat {f}}_{_{\text{IO}}}\,} &+&i\ {\widehat {f}}_{_{\text{IE}}}&+&{\widehat {f}}_{_{\text{RO}}}\end{array}}}"> From this, various relationships are apparent, for example: <ul><li>The transform of a real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{_{RE}}+f_{_{RO}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>O</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{_{RE}}+f_{_{RO}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550b87215e740c6433a22ccc7c41b0bd5a93d1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.784ex; height:3.009ex;" alt="{\displaystyle (f_{_{RE}}+f_{_{RO}})}"> is the <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">conjugate symmetric</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{RE}+i\ {\hat {f}}_{IO}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>O</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{RE}+i\ {\hat {f}}_{IO}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f91aa65958af710115cb69126789a4ee0180cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}_{RE}+i\ {\hat {f}}_{IO}.}"> Conversely, a conjugate symmetric transform implies a real-valued time-domain.</li> <li>The transform of an imaginary-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\ f_{_{IE}}+i\ f_{_{IO}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>E</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>O</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\ f_{_{IE}}+i\ f_{_{IO}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45bd7e42a412c64a9b3dc184521191442fb50115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.87ex; height:3.009ex;" alt="{\displaystyle (i\ f_{_{IE}}+i\ f_{_{IO}})}"> is the <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">conjugate antisymmetric</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{RO}+i\ {\hat {f}}_{IE},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>O</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>E</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{RO}+i\ {\hat {f}}_{IE},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5ea7847f5759a1e8d1d21b267d789d15551252" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}_{RO}+i\ {\hat {f}}_{IE},}"> and the converse is true.</li> <li>The transform of a <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">conjugate symmetric</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{_{RE}}+i\ f_{_{IO}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>O</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{_{RE}}+i\ f_{_{IO}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090dc90c3d6f9381c4d50a4b84e789aa3c7b9013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.827ex; height:3.009ex;" alt="{\displaystyle (f_{_{RE}}+i\ f_{_{IO}})}"> is the real-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{RE}+{\hat {f}}_{RO},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>E</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>O</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{RE}+{\hat {f}}_{RO},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5383aec75a13a1d41e41fd766a31c7e759033728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.354ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}_{RE}+{\hat {f}}_{RO},}"> and the converse is true.</li> <li>The transform of a <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">conjugate antisymmetric</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{_{RO}}+i\ f_{_{IE}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>O</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mtext> </mtext> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>E</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{_{RO}}+i\ f_{_{IE}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09859ebf176c9b0e5be2c875d01e628a7758c12d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.827ex; height:3.009ex;" alt="{\displaystyle (f_{_{RO}}+i\ f_{_{IE}})}"> is the imaginary-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\ {\hat {f}}_{IE}+i{\hat {f}}_{IO},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>E</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>O</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\ {\hat {f}}_{IE}+i{\hat {f}}_{IO},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13febf83a4468aac73322704ff6e84af6690153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.702ex; height:3.343ex;" alt="{\displaystyle i\ {\hat {f}}_{IE}+i{\hat {f}}_{IO},}"> and the converse is true.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Conjugation">Conjugation</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=20" title="Edit section: Conjugation">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}f(x){\bigr )}^{*}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ \left({\widehat {f}}(-\xi )\right)^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}f(x){\bigr )}^{*}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ \left({\widehat {f}}(-\xi )\right)^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51eaa8208ada744a964ebea4c4390e86ef0d3c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.489ex; height:5.176ex;" alt="{\displaystyle {\bigl (}f(x){\bigr )}^{*}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ \left({\widehat {f}}(-\xi )\right)^{*}}"> (Note: the ∗ denotes <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugation</a>.) In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is real, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> is <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">even symmetric</a> (aka <a href="/wiki/Hermitian_function" title="Hermitian function">Hermitian function</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\xi )={\bigl (}{\widehat {f}}(\xi ){\bigr )}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\xi )={\bigl (}{\widehat {f}}(\xi ){\bigr )}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa277656f3014e78e248761d97e96314761cf79c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.958ex; height:3.676ex;" alt="{\displaystyle {\widehat {f}}(-\xi )={\bigl (}{\widehat {f}}(\xi ){\bigr )}^{*}.}"> And if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is purely imaginary, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> is <a href="/wiki/Even_and_odd_functions#Complex-valued_functions" title="Even and odd functions">odd symmetric</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\xi )=-({\widehat {f}}(\xi ))^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\xi )=-({\widehat {f}}(\xi ))^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1299ca0ecd82a4ebc38c9d13e11b40e2e665fa1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.446ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(-\xi )=-({\widehat {f}}(\xi ))^{*}.}"> <div class="mw-heading mw-heading4"><h4 id="Real_and_imaginary_part_in_time">Real and imaginary part in time</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=21" title="Edit section: Real and imaginary part in time">edit</a>]</div> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2}}\left({\widehat {f}}(\xi )+{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2}}\left({\widehat {f}}(\xi )+{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd817ebefa9e41666a9d313dd2f91a8aed6b564" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.999ex; height:5.176ex;" alt="{\displaystyle \operatorname {Re} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2}}\left({\widehat {f}}(\xi )+{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2i}}\left({\widehat {f}}(\xi )-{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2i}}\left({\widehat {f}}(\xi )-{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a17e8f7015b8df5d26463189604fdcb1b9e0a98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.599ex; height:5.176ex;" alt="{\displaystyle \operatorname {Im} \{f(x)\}\ \ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ {\tfrac {1}{2i}}\left({\widehat {f}}(\xi )-{\bigl (}{\widehat {f}}(-\xi ){\bigr )}^{*}\right)}"> <div class="mw-heading mw-heading4"><h4 id="Zero_frequency_component">Zero frequency component</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=22" title="Edit section: Zero frequency component">edit</a>]</div> Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5354e193004a0e2f16e7d4a76ea499ffcca225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.291ex; height:2.509ex;" alt="{\displaystyle \xi =0}"> in the definition, we obtain: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512d3e27c203a3eadf508b01e54cc3494786bd2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.672ex; height:6.009ex;" alt="{\displaystyle {\widehat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}"> The integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> over its domain is known as the average value or <a href="/wiki/DC_bias" title="DC bias">DC bias</a> of the function. <div class="mw-heading mw-heading3"><h3 id="Invertibility_and_periodicity">Invertibility and periodicity</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=23" title="Edit section: Invertibility and periodicity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a> and <a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></div> Under suitable conditions on the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, it can be recovered from its Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}">. Indeed, denoting the Fourier transform operator by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}f:={\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>f</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}f:={\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff40f194cee63574a5ba8dde9eba123f660a991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.65ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}f:={\hat {f}}}">, then for suitable functions, applying the Fourier transform twice simply flips the function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {F}}^{2}f\right)(x)=f(-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {F}}^{2}f\right)(x)=f(-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28af50c062873b13d4bc35fd6819211a074fb026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.316ex; height:3.343ex;" alt="{\displaystyle \left({\mathcal {F}}^{2}f\right)(x)=f(-x)}">, which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{4}(f)=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{4}(f)=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f00cd4b4467bedcf9b7bac2806e5652f25489c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.523ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}^{4}(f)=f}">, so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{3}\left({\hat {f}}\right)=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{3}\left({\hat {f}}\right)=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b5be1345ccd7f17d95f2a0391c857bdc0682a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.296ex; height:4.843ex;" alt="{\displaystyle {\mathcal {F}}^{3}\left({\hat {f}}\right)=f}">. In particular the Fourier transform is invertible (under suitable conditions). More precisely, defining the parity operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {P}}f)(x)=f(-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {P}}f)(x)=f(-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833b26fbf0d49261634ba577773d45d07d9f7394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.255ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {P}}f)(x)=f(-x)}">, we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab7ea5949bf8494cf70c4d81e62e977c0c7abfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:29.737ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}"> These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of the <a href="/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a>. This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the <a href="/wiki/Time%E2%80%93frequency_domain" class="mw-redirect" title="Time–frequency domain">time–frequency domain</a> (considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to the <a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a>, which involves rotations by other angles. This can be further generalized to <a href="/wiki/Linear_canonical_transformation" title="Linear canonical transformation">linear canonical transformations</a>, which can be visualized as the action of the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <a href="/wiki/SL2(R)" title="SL2(R)">SL2(R)</a> on the time–frequency plane, with the preserved symplectic form corresponding to the <a href="#Uncertainty_principle">uncertainty principle</a>, below. This approach is particularly studied in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, under <a href="/wiki/Time%E2%80%93frequency_analysis" title="Time–frequency analysis">time–frequency analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Units">Units</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=24" title="Edit section: Units">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Spectral_density#Units" title="Spectral density">Spectral density § Units</a></div> The frequency variable must have inverse units to the units of the original function's domain (typically named <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">). For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> is measured in seconds, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> should be in cycles per second or <a href="/wiki/Hertz" title="Hertz">hertz</a>. If the scale of time is in units of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> seconds, then another Greek letter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is typically used instead to represent <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7a4c7c58f489ad173cd8e39e6135e42a2dd5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.069ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi \xi }">) in units of <a href="/wiki/Radian" title="Radian">radians</a> per second. If using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> for units of length, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> must be in inverse length, e.g., <a href="/wiki/Wavenumber" title="Wavenumber">wavenumbers</a>. That is to say, there are two versions of the real line: one which is the <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> and measured in units of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}"> and the other which is the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> and measured in inverse units to the units of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e6cc375ac6123d2342be53eba87b92fbbacf07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.486ex; height:2.009ex;" alt="{\displaystyle t.}"> These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition. In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> must always be taken to be a <a href="/wiki/Linear_form" title="Linear form">linear form</a> on the space of its domain, which is to say that the second real line is the <a href="/wiki/Dual_space" title="Dual space">dual space</a> of the first real line. See the article on <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to general <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a>, including the case of Fourier series. That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants. In other conventions, the Fourier transform has i in the exponent instead of −i, and vice versa for the inversion formula. This convention is common in modern physics<a href="#cite_note-19">[15]</a> and is the default for <a rel="nofollow" class="external text" href="https://www.wolframalpha.com">Wolfram Alpha</a>, and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c12179a82497132d68a556a537ef57b46cbbf83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.538ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )}"> is the amplitude of the wave  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i2\pi \xi x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i2\pi \xi x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fb2df821c09da15777d0d410d97ba5186d63ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.676ex;" alt="{\displaystyle e^{-i2\pi \xi x}}">  instead of the wave  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi \xi x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi \xi x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21b0e3490934b365ffe026bd724c2a029a3bc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle e^{i2\pi \xi x}}"> (the former, with its minus sign, is often seen in the time dependence for <a href="/wiki/Sinusoidal_plane-wave_solutions_of_the_electromagnetic_wave_equation" title="Sinusoidal plane-wave solutions of the electromagnetic wave equation">Sinusoidal plane-wave solutions of the electromagnetic wave equation</a>, or in the <a href="/wiki/Wave_function#Time_dependence" title="Wave function">time dependence for quantum wave functions</a>). Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve i have it replaced by −i. In <a href="/wiki/Electrical_engineering" title="Electrical engineering">Electrical engineering</a> the letter j is typically used for the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> instead of i because i is used for current. When using <a href="/wiki/Dimensionless_units" class="mw-redirect" title="Dimensionless units">dimensionless units</a>, the constant factors might not even be written in the transform definition. For instance, in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the characteristic function Φ of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential, and since the units of x are ignored, there is no 2π either: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>λ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8fb518c08834ea08d878d225aa4d7a4d683206e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.261ex; height:6.009ex;" alt="{\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}"> (In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>, i.e., measures which possess "atoms".) From the higher point of view of <a href="/wiki/Character_theory" title="Character theory">group characters</a>, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a <a href="/wiki/Locally_compact_abelian_group" title="Locally compact abelian group">locally compact Abelian group</a>. <div class="mw-heading mw-heading3"><h3 id="Uniform_continuity_and_the_Riemann–Lebesgue_lemma">Uniform continuity and the Riemann–Lebesgue lemma</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=25" title="Edit section: Uniform continuity and the Riemann–Lebesgue lemma">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rectangular_function.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Rectangular_function.svg/220px-Rectangular_function.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Rectangular_function.svg/330px-Rectangular_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Rectangular_function.svg/440px-Rectangular_function.svg.png 2x" data-file-width="840" data-file-height="580" /></a><figcaption>The <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> is <a href="/wiki/Lebesgue_integrable" class="mw-redirect" title="Lebesgue integrable">Lebesgue integrable</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sinc_function_(normalized).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Sinc_function_%28normalized%29.svg/220px-Sinc_function_%28normalized%29.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Sinc_function_%28normalized%29.svg/330px-Sinc_function_%28normalized%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Sinc_function_%28normalized%29.svg/440px-Sinc_function_%28normalized%29.svg.png 2x" data-file-width="908" data-file-height="623" /></a><figcaption>The <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.</figcaption></figure> The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties. The Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"> of any integrable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is <a href="/wiki/Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a> and<a href="#cite_note-Katznelson-1976-20">[16]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≤</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mi>f</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5a78ed502d4cfaa27fc2c44ca907bd8efb4c3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:13.656ex; height:4.009ex;" alt="{\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}"> By the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>,<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353b052333a590ecb5efc7456839e24200de5d4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.463ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}"> However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"> need not be integrable. For example, the Fourier transform of the <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a>, which is integrable, is the <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>, which is not <a href="/wiki/Lebesgue_integrable" class="mw-redirect" title="Lebesgue integrable">Lebesgue integrable</a>, because its <a href="/wiki/Improper_integral" title="Improper integral">improper integrals</a> behave analogously to the <a href="/wiki/Alternating_harmonic_series" class="mw-redirect" title="Alternating harmonic series">alternating harmonic series</a>, in converging to a sum without being <a href="/wiki/Absolutely_convergent" class="mw-redirect" title="Absolutely convergent">absolutely convergent</a>. It is not generally possible to write the inverse transform as a <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>. However, when both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"> are integrable, the inverse equality <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3dd00a6d72376d70477581d1106bbf7dd4388e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.837ex; height:6.009ex;" alt="{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{i2\pi x\xi }\,d\xi }"> holds for almost every x. As a result, the Fourier transform is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> on <a href="/wiki/Lp_space" title="Lp space">L1(R)</a>. <div class="mw-heading mw-heading3"><h3 id="Plancherel_theorem_and_Parseval's_theorem">Plancherel theorem and Parseval's theorem</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=26" title="Edit section: Plancherel theorem and Parseval's theorem">edit</a>]</div> <blockquote>Main page: <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></blockquote>Let f(x) and g(x) be integrable, and let f̂(ξ) and ĝ(ξ) be their Fourier transforms. If f(x) and g(x) are also <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a>, then the Parseval formula follows:<a href="#cite_note-21">[17]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a1ae32e4420e33a008dda1e017a54d3b898486" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.009ex; height:6.009ex;" alt="{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}"> where the bar denotes <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>. The <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a>, which follows from the above, states that<a href="#cite_note-22">[18]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4c8d69cb1b163d5b0d80b7033eef1f489fc2cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.515ex; height:6.009ex;" alt="{\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}"> Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a> on L2(R). On L1(R) ∩ L2(R), this extension agrees with original Fourier transform defined on L1(R), thus enlarging the domain of the Fourier transform to L1(R) + L2(R) (and consequently to Lp(R) for 1 ≤ p ≤ 2). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the <a href="/wiki/Energy" title="Energy">energy</a> of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem. See <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a> for a general formulation of this concept in the context of locally compact abelian groups. <div class="mw-heading mw-heading3"><h3 id="Poisson_summation_formula">Poisson summation formula</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=27" title="Edit section: Poisson summation formula">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a></div> The Poisson summation formula (PSF) is an equation that relates the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> coefficients of the <a href="/wiki/Periodic_summation" title="Periodic summation">periodic summation</a> of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions f, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7224b0df29ab2c3c029990ae25252e027854809f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.615ex; height:5.509ex;" alt="{\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}"> It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The formula has applications in engineering, physics, and <a href="/wiki/Number_theory" title="Number theory">number theory</a>. The frequency-domain dual of the standard Poisson summation formula is also called the <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a>. Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a <a href="/wiki/Theta_function" title="Theta function">theta function</a>. It is used in <a href="/wiki/Number_theory" title="Number theory">number theory</a> to prove the transformation properties of theta functions, which turn out to be a type of <a href="/wiki/Modular_form" title="Modular form">modular form</a>, and it is connected more generally to the theory of <a href="/wiki/Automorphic_form" title="Automorphic form">automorphic forms</a> where it appears on one side of the <a href="/wiki/Selberg_trace_formula" title="Selberg trace formula">Selberg trace formula</a>. <div class="mw-heading mw-heading3"><h3 id="Differentiation">Differentiation</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=28" title="Edit section: Differentiation">edit</a>]</div> Suppose f(x) is an absolutely continuous differentiable function, and both f and its derivative f′ are integrable. Then the Fourier transform of the derivative is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f42cb58f48fcb4f12168206617bec538890dc34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.54ex; height:6.176ex;" alt="{\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=i2\pi \xi {\hat {f}}(\xi ).}"> More generally, the Fourier transformation of the nth derivative f(n) is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c617ed9a74c9681a8ebd7427f113448435ab78c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.212ex; height:6.176ex;" alt="{\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(i2\pi \xi )^{n}{\hat {f}}(\xi ).}"> Analogously, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\left\{{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi )\right\}=(i2\pi x)^{n}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\left\{{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi )\right\}=(i2\pi x)^{n}f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378027fd0425903e85378560939c9c40c3ed1734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.813ex; height:6.176ex;" alt="{\displaystyle {\mathcal {F}}\left\{{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi )\right\}=(i2\pi x)^{n}f(x)}">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948d3b4963685cb698cda569634e1cb47bfabc72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.162ex; height:6.176ex;" alt="{\displaystyle {\mathcal {F}}\left\{x^{n}f(x)\right\}=\left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}}{d\xi ^{n}}}{\hat {f}}(\xi ).}"> By applying the Fourier transform and using these formulas, some <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "f(x) is smooth <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> f̂(ξ) quickly falls to 0 for |ξ| → ∞." By using the analogous rules for the inverse Fourier transform, one can also say "f(x) quickly falls to 0 for |x| → ∞ if and only if f̂(ξ) is smooth." <div class="mw-heading mw-heading3"><h3 id="Convolution_theorem">Convolution theorem</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=29" title="Edit section: Convolution theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Convolution_theorem" title="Convolution theorem">Convolution theorem</a></div> The Fourier transform translates between <a href="/wiki/Convolution" title="Convolution">convolution</a> and multiplication of functions. If f(x) and g(x) are integrable functions with Fourier transforms f̂(ξ) and ĝ(ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms f̂(ξ) and ĝ(ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear). This means that if: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98846d3276efe8aa934969785746a7d0a854caeb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.946ex; height:6.009ex;" alt="{\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}"> where ∗ denotes the convolution operation, then: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d94210e69afa7843cfb846fdbd7d81cf5d5447b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.92ex; height:3.343ex;" alt="{\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\,{\hat {g}}(\xi ).}"> In <a href="/wiki/LTI_system_theory" class="mw-redirect" title="LTI system theory">linear time invariant (LTI) system theory</a>, it is common to interpret g(x) as the <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> of an LTI system with input f(x) and output h(x), since substituting the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">unit impulse</a> for f(x) yields h(x) = g(x). In this case, ĝ(ξ) represents the <a href="/wiki/Frequency_response" title="Frequency response">frequency response</a> of the system. Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms p̂(ξ) and q̂(ξ). <div class="mw-heading mw-heading3"><h3 id="Cross-correlation_theorem">Cross-correlation theorem</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=30" title="Edit section: Cross-correlation theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation</a> and <a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin_theorem</a></div> In an analogous manner, it can be shown that if h(x) is the <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a> of f(x) and g(x): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋆</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534d8aff2c110a1aea481ecd85ec0aeda44359bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.414ex; height:6.009ex;" alt="{\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}"> then the Fourier transform of h(x) is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b38fea6ae5129649f926e7d4e53da21d43f6a3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.035ex; height:4.176ex;" alt="{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\,{\hat {g}}(\xi ).}"> As a special case, the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> of function f(x) is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋆</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489182638602996366a4c1e71b523cf5639e6c4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.739ex; height:6.009ex;" alt="{\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}"> for which <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1a1e654ca57c995d4ef5c53fbbed3bb38f0ded" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.1ex; height:4.676ex;" alt="{\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}"> <div class="mw-heading mw-heading3"><h3 id="Eigenfunctions">Eigenfunctions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=31" title="Edit section: Eigenfunctions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mehler_kernel" title="Mehler kernel">Mehler kernel</a> and <a href="/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform" title="Hermite polynomials">Hermite polynomials § Hermite functions as eigenfunctions of the Fourier transform</a></div> The Fourier transform is a linear transform which has eigenfunctions obeying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ψ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>λ</mi> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb312a1e0c7fe50ae6f17f3b157e13671bdd307e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.347ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}[\psi ]=\lambda \psi ,}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {C} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {C} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ecf7e833a4d086969a19cc5ca8e8ecdb35f517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.521ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {C} .}"> A set of eigenfunctions is found by noting that the homogeneous differential equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdeac78597b20e0cc63dafce13269df2332ca31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.82ex; height:6.176ex;" alt="{\displaystyle \left[U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)+U(x)\right]\psi (x)=0}"> leads to eigenfunctions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"> of the Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> as long as the form of the equation remains invariant under Fourier transform.<a href="#cite_note-23">[note 5]</a> In other words, every solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"> and its Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\psi }}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\psi }}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4167cc0bec54180d942279048e0d3260f8e45bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.435ex; height:3.343ex;" alt="{\displaystyle {\hat {\psi }}(\xi )}"> obey the same equation. Assuming <a href="/wiki/Ordinary_differential_equation#Existence_and_uniqueness_of_solutions" title="Ordinary differential equation">uniqueness</a> of the solutions, every solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}"> must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d626d3a1e65c94535c811c73fa83389cfb76683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle U(x)}"> can be expanded in a power series in which for all terms the same factor of either one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1,\pm i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1,\pm i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0b1c6c9b0d9c23f32d1a856110c08c9be255c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.615ex; height:2.509ex;" alt="{\displaystyle \pm 1,\pm i}"> arises from the factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f830f9a31861a4ce907be6802000083ac31dc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.021ex; height:2.343ex;" alt="{\displaystyle i^{n}}"> introduced by the <a href="#Differentiation">differentiation</a> rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e25f3a96880931a1e003f3331c30d83eb0b758b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.35ex; height:2.843ex;" alt="{\displaystyle U(x)=x}"> leads to the <a href="/wiki/Normal_distribution#Fourier_transform_and_characteristic_function" title="Normal distribution">standard normal distribution</a>.<a href="#cite_note-24">[19]</a> More generally, a set of eigenfunctions is also found by noting that the <a href="#Differentiation">differentiation</a> rules imply that the <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d064e51b07fbf1e010b2aacf43bf894c516e0c92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.382ex; height:6.176ex;" alt="{\displaystyle \left[W\left({\frac {i}{2\pi }}{\frac {d}{dx}}\right)+W(x)\right]\psi (x)=C\psi (x)}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> constant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ead36b42ec68c542b267f9e6bb62cf911a764b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.574ex; height:2.843ex;" alt="{\displaystyle W(x)}"> being a non-constant even function remains invariant in form when applying the Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> to both sides of the equation. The simplest example is provided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(x)=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(x)=x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b23988b78e48d2d08b0386e423ded47e4380981" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.057ex; height:3.176ex;" alt="{\displaystyle W(x)=x^{2}}"> which is equivalent to considering the Schrödinger equation for the <a href="/wiki/Quantum_harmonic_oscillator#Natural_length_and_energy_scales" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>.<a href="#cite_note-25">[20]</a> The corresponding solutions provide an important choice of an orthonormal basis for <a href="/wiki/Square-integrable_function" title="Square-integrable function">L2(R)</a> and are given by the "physicist's" <a href="/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform" title="Hermite polynomials">Hermite functions</a>. Equivalently one may use <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> <msqrt> <mi>n</mi> <mo>!</mo> </msqrt> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9121f54a3fdbb0eedecf2aef5a379bdfae414b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.395ex; height:6.843ex;" alt="{\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}"> where Hen(x) are the "probabilist's" <a href="/wiki/Hermite_polynomial" class="mw-redirect" title="Hermite polynomial">Hermite polynomials</a>, defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f936217fee607ce3d6ce393246bf9e1541316a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.358ex; height:6.176ex;" alt="{\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{{\frac {1}{2}}x^{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {1}{2}}x^{2}}.}"> Under this convention for the Fourier transform, we have that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095d2a5cb5865fbd3a1898cc90d47be841be4f99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.608ex; height:3.343ex;" alt="{\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi ).}"> In other words, the Hermite functions form a complete <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> system of <a href="/wiki/Eigenfunctions" class="mw-redirect" title="Eigenfunctions">eigenfunctions</a> for the Fourier transform on L2(R).<a href="#cite_note-Pinsky-2002-18">[14]</a><a href="#cite_note-26">[21]</a> However, this choice of eigenfunctions is not unique. Because of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{4}=\mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{4}=\mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175ad9d60cb40cf229dda2e01a3cd08fbf924158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.095ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}^{4}=\mathrm {id} }"> there are only four different <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the Fourier transform (the fourth roots of unity ±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction.<a href="#cite_note-27">[22]</a> As a consequence of this, it is possible to decompose L2(R) as a direct sum of four spaces H0, H1, H2, and H3 where the Fourier transform acts on Hek simply by multiplication by ik. Since the complete set of Hermite functions ψn provides a resolution of the identity they diagonalize the Fourier operator, i.e. the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eada95a2763cb70e97d43afd2f1f557fd2f046b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.03ex; height:6.509ex;" alt="{\displaystyle {\mathcal {F}}[f](\xi )=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(\xi )~.}"> This approach to define the Fourier transform was first proposed by <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>.<a href="#cite_note-Duoandikoetxea-2001-28">[23]</a> Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the <a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a> used in time–frequency analysis.<a href="#cite_note-Boashash-2003-29">[24]</a> In <a href="/wiki/Physics" title="Physics">physics</a>, this transform was introduced by <a href="/wiki/Edward_Condon" title="Edward Condon">Edward Condon</a>.<a href="#cite_note-30">[25]</a> This change of basis functions becomes possible because the Fourier transform is a unitary transform when using the right <a href="#Other_conventions">conventions</a>. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> via<a href="#cite_note-31">[26]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>ψ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>t</mi> <mi>N</mi> </mrow> </msup> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271ffa7285358aed34c7ccbdadcc08d3da01dd90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.207ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}[\psi ]=e^{-itN}\psi .}"> The operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is the <a href="/wiki/Quantum_harmonic_oscillator#Ladder_operator_method" title="Quantum harmonic oscillator">number operator</a> of the quantum harmonic oscillator written as<a href="#cite_note-auto-32">[27]</a><a href="#cite_note-33">[28]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e88b9540ce58cff6b8ecc677b2c7508bbc37ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.598ex; height:6.343ex;" alt="{\displaystyle N\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right).}"> It can be interpreted as the <a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">generator</a> of <a href="/wiki/Mehler_kernel#Fractional_Fourier_transform" title="Mehler kernel">fractional Fourier transforms</a> for arbitrary values of t, and of the conventional continuous Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> for the particular value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\pi /2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=\pi /2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d8d37287242db7fc4f83c16bbfae81379fd692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.242ex; height:2.843ex;" alt="{\displaystyle t=\pi /2,}"> with the <a href="/wiki/Mehler_kernel#Physics_version" title="Mehler kernel">Mehler kernel</a> implementing the corresponding <a href="/wiki/Active_and_passive_transformation#In_abstract_vector_spaces" title="Active and passive transformation">active transform</a>. The eigenfunctions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> are the <a href="/wiki/Hermite_polynomials#Hermite_functions" title="Hermite polynomials">Hermite functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c06a3b81927b2e45c3f68630083c5c2d9626d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \psi _{n}(x)}"> which are therefore also eigenfunctions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1656ae73ede684468b360e948a8a38e6e2c461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.573ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}.}"> Upon extending the Fourier transform to <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> the <a href="/wiki/Dirac_comb#Fourier_transform" title="Dirac comb">Dirac comb</a> is also an eigenfunction of the Fourier transform. <div class="mw-heading mw-heading3"><h3 id="Connection_with_the_Heisenberg_group">Connection with the Heisenberg group</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=32" title="Edit section: Connection with the Heisenberg group">edit</a>]</div> The <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</a> is a certain <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operators</a> on the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> L2(R) of square integrable complex valued functions f on the real line, generated by the translations (Ty f)(x) = f (x + y) and multiplication by ei2πξx, (Mξ f)(x) = ei2πξx f (x). These operators do not commute, as their (group) commutator is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{i2\pi \xi y}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ξ</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>y</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{i2\pi \xi y}f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aeeb609dc9677398e748ac8600d5d08cb5c9cfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.795ex; height:4.843ex;" alt="{\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{i2\pi \xi y}f(x)}"> which is multiplication by the constant (independent of x) ei2πξy ∈ U(1) (the <a href="/wiki/Circle_group" title="Circle group">circle group</a> of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of triples (x, ξ, z) ∈ R2 × U(1), with the group law <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{i2\pi \left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{i2\pi \left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0872cfb68f46051a588261f4178ff3b2470b85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.946ex; height:4.843ex;" alt="{\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{i2\pi \left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}"> Denote the Heisenberg group by H1. The above procedure describes not only the group structure, but also a standard <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a> of H1 on a Hilbert space, which we denote by ρ : H1 → B(L2(R)). Define the linear automorphism of R2 by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>ξ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>ξ</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e889cc64538a6abe0aa9e795dd76bceb7ef5298c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.083ex; height:6.176ex;" alt="{\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}"> so that J2 = −I. This J can be extended to a unique automorphism of H1: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-i2\pi \xi x}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>ξ</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-i2\pi \xi x}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac55fa24013137da9820eb768639a21827ea12f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:28.38ex; height:3.343ex;" alt="{\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-i2\pi \xi x}\right).}"> According to the <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>, the unitary representations ρ and ρ ∘ j are unitarily equivalent, so there is a unique intertwiner W ∈ U(L2(R)) such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \circ j=W\rho W^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>∘</mo> <mi>j</mi> <mo>=</mo> <mi>W</mi> <mi>ρ</mi> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \circ j=W\rho W^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7bf392905b0d5cf9902075b6bc17cd576e5105" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.299ex; height:2.843ex;" alt="{\displaystyle \rho \circ j=W\rho W^{*}.}"> This operator W is the Fourier transform. Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.<a href="#cite_note-34">[29]</a> For example, the square of the Fourier transform, W2, is an intertwiner associated with J2 = −I, and so we have (W2f)(x) = f (−x) is the reflection of the original function f. <div class="mw-heading mw-heading2"><h2 id="Complex_domain">Complex domain</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=33" title="Edit section: Complex domain">edit</a>]</div> The <a href="/wiki/Integral" title="Integral">integral</a> for the Fourier transform <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8d983c90ce3311b6f64ab3ba03471ab21fbe0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.088ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}"> can be studied for <a href="/wiki/Complex_number" title="Complex number">complex</a> values of its argument ξ. Depending on the properties of f, this might not converge off the real axis at all, or it might converge to a <a href="/wiki/Complex_analysis" title="Complex analysis">complex</a> <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> for all values of ξ = σ + iτ, or something in between.<a href="#cite_note-35">[30]</a> The <a href="/wiki/Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorem</a> says that f is smooth (i.e., n-times differentiable for all positive integers n) and compactly supported if and only if f̂ (σ + iτ) is a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a> for which there exists a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> a > 0 such that for any <a href="/wiki/Integer" title="Integer">integer</a> n ≥ 0, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi )\right\vert \leq Ce^{a\vert \tau \vert }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo fence="false" stretchy="false">|</mo> <mi>τ</mi> <mo fence="false" stretchy="false">|</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi )\right\vert \leq Ce^{a\vert \tau \vert }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6764f97499417d2f1605b31562c5c0c78b93d8dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.899ex; height:3.843ex;" alt="{\displaystyle \left\vert \xi ^{n}{\hat {f}}(\xi )\right\vert \leq Ce^{a\vert \tau \vert }}"> for some constant C. (In this case, f is supported on [−a, a].) This can be expressed by saying that f̂ is an <a href="/wiki/Entire_function" title="Entire function">entire function</a> which is <a href="/wiki/Rapidly_decreasing" class="mw-redirect" title="Rapidly decreasing">rapidly decreasing</a> in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).<a href="#cite_note-36">[31]</a> (If f is not smooth, but only L2, the statement still holds provided n = 0.<a href="#cite_note-37">[32]</a>) The space of such functions of a <a href="/wiki/Complex_analysis" title="Complex analysis">complex variable</a> is called the Paley—Wiener space. This theorem has been generalised to semisimple <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>.<a href="#cite_note-38">[33]</a> If f is supported on the half-line t ≥ 0, then f is said to be "causal" because the <a href="/wiki/Impulse_response_function" class="mw-redirect" title="Impulse response function">impulse response function</a> of a physically realisable <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filter</a> must have this property, as no effect can precede its cause. <a href="/wiki/Raymond_Paley" title="Raymond Paley">Paley</a> and Wiener showed that then f̂ extends to a <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a> on the complex lower half-plane τ < 0 which tends to zero as τ goes to infinity.<a href="#cite_note-39">[34]</a> The converse is false and it is not known how to characterise the Fourier transform of a causal function.<a href="#cite_note-40">[35]</a> <div class="mw-heading mw-heading3"><h3 id="Laplace_transform">Laplace transform</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=34" title="Edit section: Laplace transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Laplace_transform#Fourier_transform" title="Laplace transform">Laplace transform § Fourier transform</a></div> The Fourier transform f̂(ξ) is related to the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> F(s), which is also used for the solution of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> and the analysis of <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filters</a>. It may happen that a function f for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. For example, if f(t) is of exponential growth, i.e., <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>C</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo fence="false" stretchy="false">|</mo> <mi>t</mi> <mo fence="false" stretchy="false">|</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87d33764327f151fbea74b86ccb215e5447b5a4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.78ex; height:3.343ex;" alt="{\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}"> for some constants C, a ≥ 0, then<a href="#cite_note-Kolmogorov-Fomin-1999-41">[36]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>τ</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c86638fd1d69c8521cfc2905b786006f54f5e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.985ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}"> convergent for all 2πτ < −a, is the <a href="/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a> of f. The more usual version ("one-sided") of the Laplace transform is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac408c7ea5f7799e185a4e8d66e69fb0964e2c02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.442ex; height:5.843ex;" alt="{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}"> If f is also causal, and analytical, then: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8717ea6905f51fa694be64df7f62d2e6fe17cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.313ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).}"> Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable s = i2πξ. From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements. Two particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle. Such designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb. Furthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response. There, only the delay circuit in-between admits a classical Fourier description, which is critical. Both the circuits to the side are unstable, and do not admit a convergent Fourier decomposition. However, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel. In modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Inversion">Inversion</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=35" title="Edit section: Inversion">edit</a>]</div> Still with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =\sigma +i\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =\sigma +i\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab740eeb5f591cc7f0eaa40dfa60ecf184aa6ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.303ex; height:2.509ex;" alt="{\displaystyle \xi =\sigma +i\tau }">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153a4a4d50ef7099fc5f8804e34dccc539f08743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}"> is complex analytic for a ≤ τ ≤ b, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{i2\pi \xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{i2\pi \xi t}\,d\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>a</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{i2\pi \xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{i2\pi \xi t}\,d\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5756eed77aca41be595193c4ce49a73965aec8e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.758ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{i2\pi \xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{i2\pi \xi t}\,d\sigma }"> by <a href="/wiki/Cauchy%27s_integral_theorem" title="Cauchy's integral theorem">Cauchy's integral theorem</a>. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.<a href="#cite_note-42">[37]</a> Theorem: If f(t) = 0 for t < 0, and |f(t)| < Cea|t| for some constants C, a > 0, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{i2\pi \xi t}\,d\sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{i2\pi \xi t}\,d\sigma ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50751082b13c152801259ea5bb4d8e42eec335c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.091ex; height:6.009ex;" alt="{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{i2\pi \xi t}\,d\sigma ,}"> for any τ < −<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠a/2π⁠. This theorem implies the <a href="/wiki/Inverse_Laplace_transform#Mellin's_inverse_formula" title="Inverse Laplace transform">Mellin inversion formula</a> for the Laplace transformation,<a href="#cite_note-Kolmogorov-Fomin-1999-41">[36]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\frac {1}{i2\pi }}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mi>i</mi> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>+</mo> <mi>i</mi> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {1}{i2\pi }}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e741ad150bf7e8af7f0aa3ce899b0387cd9aba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.839ex; height:6.509ex;" alt="{\displaystyle f(t)={\frac {1}{i2\pi }}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}"> for any b > a, where F(s) is the Laplace transform of f(t). The hypotheses can be weakened, as in the results of Carleson and Hunt, to f(t) e−at being L1, provided that f be of bounded variation in a closed neighborhood of t (cf. <a href="/wiki/Dini_test" title="Dini test">Dini test</a>), the value of f at t be taken to be the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.<a href="#cite_note-43">[38]</a> L2 versions of these inversion formulas are also available.<a href="#cite_note-44">[39]</a> <div class="mw-heading mw-heading2"><h2 id="Fourier_transform_on_Euclidean_space">Fourier transform on Euclidean space</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=36" title="Edit section: Fourier transform on Euclidean space">edit</a>]</div> The Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions. For an integrable function f(x), this article takes the definition: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a344746a58841c00190588c0d0f4a5d7a09df5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.683ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}({\boldsymbol {\xi }})={\mathcal {F}}(f)({\boldsymbol {\xi }})=\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} }"> where x and ξ are n-dimensional <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a>, and x · ξ is the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the vectors. Alternatively, ξ can be viewed as belonging to the <a href="/wiki/Dual_space" title="Dual space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n\star }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>⋆</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n\star }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cca3797ce87e7fcb30722dd571fb8adb0e278a2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.719ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n\star }}">, in which case the dot product becomes the <a href="/wiki/Tensor_contraction" title="Tensor contraction">contraction</a> of x and ξ, usually written as ⟨x, ξ⟩. All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a> holds.<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Uncertainty_principle">Uncertainty principle</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=37" title="Edit section: Uncertainty principle">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a></div> Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform f̂(ξ) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform. The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> by viewing a function and its Fourier transform as <a href="/wiki/Conjugate_variables" title="Conjugate variables">conjugate variables</a> with respect to the <a href="/wiki/Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a> on the <a href="/wiki/Time%E2%80%93frequency_representation" title="Time–frequency representation">time–frequency domain</a>: from the point of view of the <a href="/wiki/Linear_canonical_transformation" title="Linear canonical transformation">linear canonical transformation</a>, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the <a href="/wiki/Symplectic_vector_space" title="Symplectic vector space">symplectic form</a>. Suppose f(x) is an integrable and <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> function. Without loss of generality, assume that f(x) is normalized: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412ed9826b5feef23ab814a47116c40a73c03654" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.439ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}"> It follows from the <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> that f̂(ξ) is also normalized. The spread around x = 0 may be measured by the dispersion about zero<a href="#cite_note-45">[40]</a> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd8971d09bc6b8d2ef7830ef77554bb4c1075b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.727ex; height:6.009ex;" alt="{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}"> In probability terms, this is the <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">second moment</a> of |f(x)|2 about zero. The uncertainty principle states that, if f(x) is absolutely continuous and the functions x·f(x) and f′(x) are square integrable, then<a href="#cite_note-Pinsky-2002-18">[14]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6503aa600c5cfe1925fea1161d9bed941b9c186" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.848ex; height:5.509ex;" alt="{\displaystyle D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}.}"> The equality is attained only in the case <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>∴</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>σ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1509eaf1024b829e9dec04924b690b296545930" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.888ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}"> where σ > 0 is arbitrary and C1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4√2/√σ⁠ so that f is L2-normalized.<a href="#cite_note-Pinsky-2002-18">[14]</a> In other words, where f is a (normalized) <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a> with variance σ2/2π, centered at zero, and its Fourier transform is a Gaussian function with variance σ−2/2π. In fact, this inequality implies that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e275f58314e6f2dae0b56d7b41f4e047241c6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.212ex; height:6.176ex;" alt="{\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}}"> for any x0, ξ0 ∈ R.<a href="#cite_note-Stein-Shakarchi-2003-46">[41]</a> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the <a href="/wiki/Momentum" title="Momentum">momentum</a> and position <a href="/wiki/Wave_function" title="Wave function">wave functions</a> are Fourier transform pairs, up to a factor of the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. With this constant properly taken into account, the inequality above becomes the statement of the <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>.<a href="#cite_note-47">[42]</a> A stronger uncertainty principle is the <a href="/wiki/Hirschman_uncertainty" class="mw-redirect" title="Hirschman uncertainty">Hirschman uncertainty principle</a>, which is expressed as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mi>f</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>≥</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48da72dddb524aa18ebd5112150fe9c2eabb0ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.811ex; height:4.843ex;" alt="{\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}"> where H(p) is the <a href="/wiki/Differential_entropy" title="Differential entropy">differential entropy</a> of the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> p(x): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f76d04f1462438452951fe9a983fcc2291b94c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.595ex; height:6.009ex;" alt="{\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}"> where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case. <div class="mw-heading mw-heading3"><h3 id="Sine_and_cosine_transforms">Sine and cosine transforms</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=38" title="Edit section: Sine and cosine transforms">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sine_and_cosine_transforms" title="Sine and cosine transforms">Sine and cosine transforms</a></div> Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically<a href="#cite_note-48">[43]</a>) λ by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999af366b1feb6c82c8fe06b12aa4d323700514e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.622ex; height:5.843ex;" alt="{\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}"> This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2f2e8ba2f6913c2694b0577f5b05435ee802f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.243ex; height:6.009ex;" alt="{\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f6c23738fd13cc35b95e242c291f2790855053" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.402ex; height:6.009ex;" alt="{\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}"> Older literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b. The function f can be recovered from the sine and cosine transform using <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ed114828d7d53a20ce083a47a2ef22d6c5fbcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.392ex; height:6.009ex;" alt="{\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}"> together with trigonometric identities. This is referred to as Fourier's integral formula.<a href="#cite_note-Kolmogorov-Fomin-1999-41">[36]</a><a href="#cite_note-49">[44]</a><a href="#cite_note-50">[45]</a><a href="#cite_note-51">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Spherical_harmonics">Spherical harmonics</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=39" title="Edit section: Spherical harmonics">edit</a>]</div> Let the set of <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous</a> <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic</a> <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> of degree k on Rn be denoted by Ak. The set Ak consists of the <a href="/wiki/Solid_spherical_harmonics" class="mw-redirect" title="Solid spherical harmonics">solid spherical harmonics</a> of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e−π|x|2P(x) for some P(x) in Ak, then f̂(ξ) = i−k f(ξ). Let the set Hk be the closure in L2(Rn) of linear combinations of functions of the form f(|x|)P(x) where P(x) is in Ak. The space L2(Rn) is then a direct sum of the spaces Hk and the Fourier transform maps each space Hk to itself and is possible to characterize the action of the Fourier transform on each space Hk.<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> Let f(x) = f0(|x|)P(x) (with P(x) in Ak), then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7c5e156a6b5bfb0d393732ffb831fa32efa78e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.904ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a708c74339047ce9b557330cce02c84fa38ad9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.087ex; height:5.843ex;" alt="{\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}"> Here J(n + 2k − 2)/2 denotes the <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind with order <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n + 2k − 2/2⁠. When k = 0 this gives a useful formula for the Fourier transform of a radial function.<a href="#cite_note-52">[47]</a> This is essentially the <a href="/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a>. Moreover, there is a simple recursion relating the cases n + 2 and n<a href="#cite_note-53">[48]</a> allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one. <div class="mw-heading mw-heading3"><h3 id="Restriction_problems">Restriction problems</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=40" title="Edit section: Restriction problems">edit</a>]</div> In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L2(Rn) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in Lp for 1 < p < 2. It is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in Rn is of particular interest. In this case the Tomas–<a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein</a> restriction theorem states that the restriction of the Fourier transform to the unit sphere in Rn is a bounded operator on Lp provided 1 ≤ p ≤ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2n + 2/n + 3⁠. One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets ER indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function f, consider the function fR defined by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942e2a9fb97ac1f7109150d1908e44185b45aa95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.173ex; height:6.009ex;" alt="{\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}"> Suppose in addition that f ∈ Lp(Rn). For n = 1 and 1 < p < ∞, if one takes ER = (−R, R), then fR converges to f in Lp as R tends to infinity, by the boundedness of the <a href="/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a>. Naively one may hope the same holds true for n > 1. In the case that ER is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball ER = {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in Lp(Rn). For n ≥ 2 it is a celebrated theorem of <a href="/wiki/Charles_Fefferman" title="Charles Fefferman">Charles Fefferman</a> that the multiplier for the unit ball is never bounded unless p = 2.<a href="#cite_note-Duoandikoetxea-2001-28">[23]</a> In fact, when p ≠ 2, this shows that not only may fR fail to converge to f in Lp, but for some functions f ∈ Lp(Rn), fR is not even an element of Lp. <div class="mw-heading mw-heading2"><h2 id="Fourier_transform_on_function_spaces">Fourier transform on function spaces</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=41" title="Edit section: Fourier transform on function spaces">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="On_Lp_spaces">On Lp spaces</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=42" title="Edit section: On Lp spaces">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="On_L1">On L1</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=43" title="Edit section: On L1">edit</a>]</div> The definition of the Fourier transform by the integral formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4646e8289845c34e138fef8ce15d6d93d21ccd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.102ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \xi \cdot x}\,dx}"> is valid for Lebesgue integrable functions f; that is, f ∈ L1(Rn). The Fourier transform F : L1(Rn) → L∞(Rn) is a <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operator</a>. This follows from the observation that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3603df5f6dccabf060829631c7ad0556a33539" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.285ex; height:5.676ex;" alt="{\displaystyle \left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}"> which shows that its <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a> is bounded by 1. Indeed, it equals 1, which can be seen, for example, from the <a href="#rect">transform of the rect function</a>. The image of L1 is a subset of the space C0(Rn) of continuous functions that tend to zero at infinity (the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>), although it is not the entire space. Indeed, there is no simple characterization of the image. <div class="mw-heading mw-heading4"><h4 id="On_L2">On L2</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=44" title="Edit section: On L2">edit</a>]</div> Since compactly supported smooth functions are integrable and dense in L2(Rn), the <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> allows one to extend the definition of the Fourier transform to general functions in L2(Rn) by continuity arguments. The Fourier transform in L2(Rn) is no longer given by an ordinary Lebesgue integral, although it can be computed by an <a href="/wiki/Improper_integral" title="Improper integral">improper integral</a>, here meaning that for an L2 function f, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-i2\pi \xi \cdot x}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>R</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-i2\pi \xi \cdot x}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b07196d1802125c7b4a2a82ca2f5f4c0dae2eef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.252ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-i2\pi \xi \cdot x}\,dx}"> where the limit is taken in the L2 sense.<a href="#cite_note-54">[49]</a><a href="#cite_note-55">[50]</a>) Many of the properties of the Fourier transform in L1 carry over to L2, by a suitable limiting argument. Furthermore, F : L2(Rn) → L2(Rn) is a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a>.<a href="#cite_note-56">[51]</a> For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any f, g ∈ L2(Rn) we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef488ab107e752cda01a5d83d80d462b630536a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.936ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}"> In particular, the image of L2(Rn) is itself under the Fourier transform. <div class="mw-heading mw-heading4"><h4 id="On_other_Lp">On other Lp</h4>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=45" title="Edit section: On other Lp">edit</a>]</div> The definition of the Fourier transform can be extended to functions in Lp(Rn) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L2 plus a fat body part in L1. In each of these spaces, the Fourier transform of a function in Lp(Rn) is in Lq(Rn), where q = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠p/p − 1⁠ is the <a href="/wiki/H%C3%B6lder_conjugate" class="mw-redirect" title="Hölder conjugate">Hölder conjugate</a> of p (by the <a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young inequality</a>). However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in Lp for the range 2 < p < ∞ requires the study of distributions.<a href="#cite_note-Katznelson-1976-20">[16]</a> In fact, it can be shown that there are functions in Lp with p > 2 so that the Fourier transform is not defined as a function.<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Tempered_distributions">Tempered distributions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=46" title="Edit section: Tempered distributions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">Distribution (mathematics) § Tempered distributions and Fourier transform</a></div> One might consider enlarging the domain of the Fourier transform from L1 + L2 by considering <a href="/wiki/Generalized_function" title="Generalized function">generalized functions</a>, or distributions. A distribution on Rn is a continuous linear functional on the space Cc(Rn) of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on Cc(Rn) and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map Cc(Rn) to Cc(Rn). In fact the Fourier transform of an element in Cc(Rn) can not vanish on an open set; see the above discussion on the uncertainty principle. The right space here is the slightly larger space of <a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz functions</a>. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> The tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support. For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let f̂ and ĝ be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,<a href="#cite_note-Stein-Weiss-1971-15">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e845d28f005056b87fad44dbe1be28aad88009" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.618ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}"> Every integrable function f defines (induces) a distribution Tf by the relation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{f}(\varphi )=\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{f}(\varphi )=\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ad9d916c0f54bf66d7a659da6bca0898cab856" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.994ex; height:5.676ex;" alt="{\displaystyle T_{f}(\varphi )=\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}"> for all Schwartz functions φ. So it makes sense to define Fourier transform T̂f of Tf by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}_{f}(\varphi )=T_{f}\left({\hat {\varphi }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}_{f}(\varphi )=T_{f}\left({\hat {\varphi }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415a074080a4176132df20ecb0c3a14b94aa763b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.563ex; height:3.509ex;" alt="{\displaystyle {\hat {T}}_{f}(\varphi )=T_{f}\left({\hat {\varphi }}\right)}"> for all Schwartz functions φ. Extending this to all tempered distributions T gives the general definition of the Fourier transform. Distributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions. <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=47" title="Edit section: Generalizations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Fourier–Stieltjes_transform">Fourier–Stieltjes transform</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=48" title="Edit section: Fourier–Stieltjes transform">edit</a>]</div> The Fourier transform of a <a href="/wiki/Finite_measure" title="Finite measure">finite</a> <a href="/wiki/Borel_measure" title="Borel measure">Borel measure</a> μ on Rn is given by:<a href="#cite_note-57">[52]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475f060402cb2377015a5d87407cdc9a6ce2d5c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.106ex; height:5.676ex;" alt="{\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu .}"> This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the <a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a> fails for measures.<a href="#cite_note-Katznelson-1976-20">[16]</a> In the case that dμ = f(x) dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a>, but the typical conventions in probability theory take eiξx instead of e−i2πξx.<a href="#cite_note-Pinsky-2002-18">[14]</a> In the case when the distribution has a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants. The Fourier transform may be used to give a characterization of measures. <a href="/wiki/Bochner%27s_theorem" title="Bochner's theorem">Bochner's theorem</a> characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.<a href="#cite_note-Katznelson-1976-20">[16]</a> Furthermore, the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, although not a function, is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used). <div class="mw-heading mw-heading3"><h3 id="Locally_compact_abelian_groups">Locally compact abelian groups</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=49" title="Edit section: Locally compact abelian groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a></div> The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> that is at the same time a <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff topological space</a> so that the group operation is continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a>. For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its <a href="/wiki/Character_group" title="Character group">characters</a>. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the <a href="/wiki/Compact-open_topology" title="Compact-open topology">compact-open topology</a> on the space of all continuous functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> to the <a href="/wiki/Circle_group" title="Circle group">circle group</a>), the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L1(G), its Fourier transform is defined by<a href="#cite_note-Katznelson-1976-20">[16]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \quad {\text{for any }}\xi \in {\hat {G}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for any </mtext> </mrow> <mi>ξ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \quad {\text{for any }}\xi \in {\hat {G}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bacdebd5d321f785d6b07a139417a957a9e631" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.728ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \quad {\text{for any }}\xi \in {\hat {G}}.}"> The Riemann–Lebesgue lemma holds in this case; f̂(ξ) is a function vanishing at infinity on Ĝ. The Fourier transform on T = R/Z is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:</mo> <mi>T</mi> <mo stretchy="false">→</mo> <mi>G</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>∣</mo> <mi>k</mi> <mo>∈</mo> <mi>Z</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b4230197b57f7677661499335a67ae501f229a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.344ex; height:2.843ex;" alt="{\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}(x)=e^{i2\pi kx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}(x)=e^{i2\pi kx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f919c131e19de33ee92a27d40ec1c0006ffbca88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.853ex; height:3.176ex;" alt="{\displaystyle e_{k}(x)=e^{i2\pi kx}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c507e969d5146c8e2343bb2bc383a3b13b1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.807ex; height:2.176ex;" alt="{\displaystyle x\in T}">. The character of such representation, that is the trace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3800dbabcd39c3c777e5306d6f1f6702015d5f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.311ex; height:2.843ex;" alt="{\displaystyle e_{k}(x)}"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c507e969d5146c8e2343bb2bc383a3b13b1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.807ex; height:2.176ex;" alt="{\displaystyle x\in T}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b31dbc35d9679701dc30d70faf1784b1a4b2dccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.732ex; height:2.176ex;" alt="{\displaystyle k\in Z}">, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i2\pi kx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i2\pi kx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/626eb811d1e0d4863c64594748669014d2364fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.676ex;" alt="{\displaystyle e^{i2\pi kx}}"> itself. In the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma. Now the group T is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3800dbabcd39c3c777e5306d6f1f6702015d5f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.311ex; height:2.843ex;" alt="{\displaystyle e_{k}(x)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in T,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0f2f879f5bba67beb1df0ba698b36c433312de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.453ex; height:2.509ex;" alt="{\displaystyle x\in T,}"> and the inner product between two class functions (all functions being class functions since T is abelian) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in L^{2}(T,d\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in L^{2}(T,d\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08a3b39c9a85ee3439964cc07060c159ec85d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.003ex; height:3.176ex;" alt="{\displaystyle f,g\in L^{2}(T,d\mu )}"> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b336a412429a159a78cdb8cdeea2f67365bb73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.573ex; height:4.176ex;" alt="{\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)}"> with the normalizing factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |T|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |T|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6adf5981e662836f3a29882b5dc49c2ff7cf9511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.191ex; height:2.843ex;" alt="{\displaystyle |T|=1}">. The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{k}\mid k\in Z\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∣</mo> <mi>k</mi> <mo>∈</mo> <mi>Z</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{k}\mid k\in Z\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513174f5f17c4ae40047a0a65f51d039472b13a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.167ex; height:2.843ex;" alt="{\displaystyle \{e_{k}\mid k\in Z\}}"> is an orthonormal basis of the space of class functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(T,d\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(T,d\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d38e4dd33334a88bde90cc837eb46c207417215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.734ex; height:3.176ex;" alt="{\displaystyle L^{2}(T,d\mu )}">. For any representation V of a finite group G, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32faa75b318d420fcea4f44307a1a25b3ebd628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}"> can be expressed as the span <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fea192dcd3b1507fdbe6982ec05499ea0ba72f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.575ex; height:3.009ex;" alt="{\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}"> are the irreps of G), such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </munder> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>χ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c29fe5aa194d10170324d295066eeb5c8f57ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.776ex; height:4.176ex;" alt="{\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}">. Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdf476974c6479fb53155364337cd48b2fe4266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.561ex; height:2.176ex;" alt="{\displaystyle G=T}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(T,d\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(T,d\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcb1a86c24f7132e7fb991027e3438c73ebab81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.853ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(T,d\mu )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>Z</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e570cfb499d1447cbe20e9d532a69d463ec4730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.622ex; height:3.509ex;" alt="{\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}">. The Pontriagin dual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37fed66c5d4e05d936ce71a8feea9a7509b8cb1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:2.843ex;" alt="{\displaystyle {\hat {T}}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{k}\}(k\in Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>∈</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{k}\}(k\in Z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85e8c4f54411af1ee568c1190c9865214f9b275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.039ex; height:2.843ex;" alt="{\displaystyle \{e_{k}\}(k\in Z)}"> and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(T,d\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(T,d\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcb1a86c24f7132e7fb991027e3438c73ebab81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.853ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(T,d\mu )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {f}}(k)={\frac {1}{|T|}}\int _{[0,1)}f(y)e^{-i2\pi ky}dy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>y</mi> </mrow> </msup> <mi>d</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\hat {f}}(k)={\frac {1}{|T|}}\int _{[0,1)}f(y)e^{-i2\pi ky}dy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53b088c267eb1450e2fc5c0115848f8d61256eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.242ex; height:4.343ex;" alt="{\textstyle {\hat {f}}(k)={\frac {1}{|T|}}\int _{[0,1)}f(y)e^{-i2\pi ky}dy}"> is its Fourier transform for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}\in {\hat {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}\in {\hat {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e705b0965259254a86dffe26390df55cc8cc8a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.787ex; height:3.176ex;" alt="{\displaystyle e_{k}\in {\hat {T}}}">. <div class="mw-heading mw-heading3"><h3 id="Gelfand_transform">Gelfand transform</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=50" title="Edit section: Gelfand transform">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></div> The Fourier transform is also a special case of <a href="/wiki/Gelfand_transform" class="mw-redirect" title="Gelfand transform">Gelfand transform</a>. In this particular context, it is closely related to the Pontryagin duality map defined above. Given an abelian <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="/wiki/Topological_group" title="Topological group">topological group</a> G, as before we consider space L1(G), defined using a Haar measure. With convolution as multiplication, L1(G) is an abelian <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a>. It also has an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> * given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54da9720f17956be1bdd293e34e56eb730f1d2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.407ex; height:4.009ex;" alt="{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.}"> Taking the completion with respect to the largest possibly C*-norm gives its enveloping C*-algebra, called the group C*-algebra C*(G) of G. (Any C*-norm on L1(G) is bounded by the L1 norm, therefore their supremum exists.) Given any abelian C*-algebra A, the Gelfand transform gives an isomorphism between A and C0(A^), where A^ is the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology. The map is simply given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto {\bigl (}\varphi \mapsto \varphi (a){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>φ</mi> <mo stretchy="false">↦</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto {\bigl (}\varphi \mapsto \varphi (a){\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4882f9cf9ce29c39fbb42ef3cc7f46b5291436db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.667ex; height:3.176ex;" alt="{\displaystyle a\mapsto {\bigl (}\varphi \mapsto \varphi (a){\bigr )}}"> It turns out that the multiplicative linear functionals of C*(G), after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset L1(G) is the Fourier–Pontryagin transform. <div class="mw-heading mw-heading3"><h3 id="Compact_non-abelian_groups">Compact non-abelian groups</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=51" title="Edit section: Compact non-abelian groups">edit</a>]</div> The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is <a href="/wiki/Compact_space" title="Compact space">compact</a>. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators.<a href="#cite_note-58">[53]</a> The Fourier transform on compact groups is a major tool in <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a><a href="#cite_note-59">[54]</a> and <a href="/wiki/Non-commutative_harmonic_analysis" class="mw-redirect" title="Non-commutative harmonic analysis">non-commutative harmonic analysis</a>. Let G be a compact <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="/wiki/Topological_group" title="Topological group">topological group</a>. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representations</a>, along with a definite choice of representation U(σ) on the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> Hσ of finite dimension dσ for each σ ∈ Σ. If μ is a finite <a href="/wiki/Borel_measure" title="Borel measure">Borel measure</a> on G, then the Fourier–Stieltjes transform of μ is the operator on Hσ defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\hat {\mu }}\xi ,\eta \right\rangle _{H_{\sigma }}=\int _{G}\left\langle {\overline {U}}_{g}^{(\sigma )}\xi ,\eta \right\rangle \,d\mu (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mrow> <mo>⟨</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>U</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> </mrow> <mo>⟩</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\hat {\mu }}\xi ,\eta \right\rangle _{H_{\sigma }}=\int _{G}\left\langle {\overline {U}}_{g}^{(\sigma )}\xi ,\eta \right\rangle \,d\mu (g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f89aa861c57a0e8bf1c5d1055eb0005ec12bb71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.785ex; height:6.176ex;" alt="{\displaystyle \left\langle {\hat {\mu }}\xi ,\eta \right\rangle _{H_{\sigma }}=\int _{G}\left\langle {\overline {U}}_{g}^{(\sigma )}\xi ,\eta \right\rangle \,d\mu (g)}"> where U(σ) is the complex-conjugate representation of U(σ) acting on Hσ. If μ is <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> with respect to the <a href="/wiki/Haar_measure" title="Haar measure">left-invariant probability measure</a> λ on G, <a href="/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">represented</a> as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu =f\,d\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>μ</mi> <mo>=</mo> <mi>f</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mu =f\,d\lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae057c70b708cff31dd3dbb1522560286fe9fc46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.953ex; height:2.676ex;" alt="{\displaystyle d\mu =f\,d\lambda }"> for some f ∈ <a href="/wiki/Lp_space" title="Lp space">L1(λ)</a>, one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ. The mapping <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \mapsto {\hat {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \mapsto {\hat {\mu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7c0a8e3150bed329d13c2fe3f35144418917d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.417ex; height:2.676ex;" alt="{\displaystyle \mu \mapsto {\hat {\mu }}}"> defines an isomorphism between the <a href="/wiki/Banach_space" title="Banach space">Banach space</a> M(G) of finite Borel measures (see <a href="/wiki/Rca_space" class="mw-redirect" title="Rca space">rca space</a>) and a closed subspace of the Banach space C∞(Σ) consisting of all sequences E = (Eσ) indexed by Σ of (bounded) linear operators Eσ : Hσ → Hσ for which the norm <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|E\|=\sup _{\sigma \in \Sigma }\left\|E_{\sigma }\right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>E</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mrow> </munder> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|E\|=\sup _{\sigma \in \Sigma }\left\|E_{\sigma }\right\|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab5e97c6d80c90b5ce3b5e8da808b2399a3f8d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.3ex; height:4.509ex;" alt="{\displaystyle \|E\|=\sup _{\sigma \in \Sigma }\left\|E_{\sigma }\right\|}"> is finite. The "<a href="/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a>" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> into a subspace of C∞(Σ). Multiplication on M(G) is given by <a href="/wiki/Convolution" title="Convolution">convolution</a> of measures and the involution * defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1bbf3b1e9a5f193f4ad7ec24f3034071700b003" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.407ex; height:4.009ex;" alt="{\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}},}"> and C∞(Σ) has a natural C*-algebra structure as Hilbert space operators. The <a href="/wiki/Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> holds, and a version of the Fourier inversion formula (<a href="/wiki/Plancherel%27s_theorem" class="mw-redirect" title="Plancherel's theorem">Plancherel's theorem</a>) follows: if f ∈ L2(G), then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} \left({\hat {f}}(\sigma )U_{g}^{(\sigma )}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mrow> </munder> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} \left({\hat {f}}(\sigma )U_{g}^{(\sigma )}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad95b47fb425be4071a4be2293e5e641f4658d9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.536ex; height:6.176ex;" alt="{\displaystyle f(g)=\sum _{\sigma \in \Sigma }d_{\sigma }\operatorname {tr} \left({\hat {f}}(\sigma )U_{g}^{(\sigma )}\right)}"> where the summation is understood as convergent in the L2 sense. The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of <a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] In this context, a categorical generalization of the Fourier transform to noncommutative groups is <a href="/wiki/Tannaka%E2%80%93Krein_duality" title="Tannaka–Krein duality">Tannaka–Krein duality</a>, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions. <div class="mw-heading mw-heading2"><h2 id="Alternatives">Alternatives</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=52" title="Edit section: Alternatives">edit</a>]</div> In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably <a href="/wiki/Transient_(acoustics)" title="Transient (acoustics)">transients</a>, or any signal of finite extent. As alternatives to the Fourier transform, in <a href="/wiki/Time%E2%80%93frequency_analysis" title="Time–frequency analysis">time–frequency analysis</a>, one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the <a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a>, <a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a>, Synchrosqueezing Fourier transform,<a href="#cite_note-60">[55]</a> or other functions to represent signals, as in <a href="/wiki/Wavelet_transform" title="Wavelet transform">wavelet transforms</a> and <a href="/wiki/Chirplet_transform" title="Chirplet transform">chirplet transforms</a>, with the wavelet analog of the (continuous) Fourier transform being the <a href="/wiki/Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a>.<a href="#cite_note-Boashash-2003-29">[24]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=53" title="Edit section: Applications">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Spectral_density#Applications" title="Spectral density">Spectral density § Applications</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg/400px-Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg.png" decoding="async" width="400" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg/600px-Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg/800px-Commutative_diagram_illustrating_problem_solving_via_the_Fourier_transform.svg.png 2x" data-file-width="308" data-file-height="66" /></a><figcaption>Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.</figcaption></figure> Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of <a href="/wiki/Derivative" title="Derivative">differentiation</a> in the time domain corresponds to multiplication by the frequency,<a href="#cite_note-61">[note 6]</a> so some <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> are easier to analyze in the frequency domain. Also, <a href="/wiki/Convolution" title="Convolution">convolution</a> in the time domain corresponds to ordinary multiplication in the frequency domain (see <a href="/wiki/Convolution_theorem" title="Convolution theorem">Convolution theorem</a>). After performing the desired operations, transformation of the result can be made back to the time domain. <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a> is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics. <div class="mw-heading mw-heading3"><h3 id="Analysis_of_differential_equations">Analysis of differential equations</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=54" title="Edit section: Analysis of differential equations">edit</a>]</div> Perhaps the most important use of the Fourier transformation is to solve <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. Many of the equations of the mathematical physics of the nineteenth century can be treated this way. Fourier studied the heat equation, which in one dimension and in dimensionless units is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial y(x,t)}{\partial t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial y(x,t)}{\partial t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62c16f4d83ef3917ed9bd39ef6ba7d4d4ab09fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.469ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial y(x,t)}{\partial t}}.}"> The example we will give, a slightly more difficult one, is the wave equation in one dimension, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial ^{2}y(x,t)}{\partial ^{2}t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial ^{2}y(x,t)}{\partial ^{2}t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e29fe6140726b3e3c9755d81c87ce36c6c2308d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.549ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial ^{2}x}}={\frac {\partial ^{2}y(x,t)}{\partial ^{2}t}}.}"> As usual, the problem is not to find a solution: there are infinitely many. The problem is that of the so-called "boundary problem": find a solution which satisfies the "boundary conditions" <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x,0)=f(x),\qquad {\frac {\partial y(x,0)}{\partial t}}=g(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,0)=f(x),\qquad {\frac {\partial y(x,0)}{\partial t}}=g(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f533141d6e5d0482eba54d0c42631ac7233e91b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.331ex; height:5.843ex;" alt="{\displaystyle y(x,0)=f(x),\qquad {\frac {\partial y(x,0)}{\partial t}}=g(x).}"> Here, f and g are given functions. For the heat equation, only one boundary condition can be required (usually the first one). But for the wave equation, there are still infinitely many solutions y which satisfy the first boundary condition. But when one imposes both conditions, there is only one possible solution. It is easier to find the Fourier transform ŷ of the solution than to find the solution directly. This is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function. After ŷ is determined, we can apply the inverse Fourier transformation to find y. Fourier's method is as follows. First, note that any function of the forms <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\bigl (}2\pi \xi (x\pm t){\bigr )}{\text{ or }}\sin {\bigl (}2\pi \xi (x\pm t){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>±</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>±</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\bigl (}2\pi \xi (x\pm t){\bigr )}{\text{ or }}\sin {\bigl (}2\pi \xi (x\pm t){\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f41fd833504333cb414206e733f9acd8413ee658" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.31ex; height:3.176ex;" alt="{\displaystyle \cos {\bigl (}2\pi \xi (x\pm t){\bigr )}{\text{ or }}\sin {\bigl (}2\pi \xi (x\pm t){\bigr )}}"> satisfies the wave equation. These are called the elementary solutions. Second, note that therefore any integral <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y(x,t)=\int _{0}^{\infty }d\xi {\Bigl [}&a_{+}(\xi )\cos {\bigl (}2\pi \xi (x+t){\bigr )}+a_{-}(\xi )\cos {\bigl (}2\pi \xi (x-t){\bigr )}+{}\\&b_{+}(\xi )\sin {\bigl (}2\pi \xi (x+t){\bigr )}+b_{-}(\xi )\sin \left(2\pi \xi (x-t)\right){\Bigr ]}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>d</mi> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y(x,t)=\int _{0}^{\infty }d\xi {\Bigl [}&a_{+}(\xi )\cos {\bigl (}2\pi \xi (x+t){\bigr )}+a_{-}(\xi )\cos {\bigl (}2\pi \xi (x-t){\bigr )}+{}\\&b_{+}(\xi )\sin {\bigl (}2\pi \xi (x+t){\bigr )}+b_{-}(\xi )\sin \left(2\pi \xi (x-t)\right){\Bigr ]}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65c9bf34f6bc5abac0d29243bda58f8c3310c2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:67.646ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}y(x,t)=\int _{0}^{\infty }d\xi {\Bigl [}&a_{+}(\xi )\cos {\bigl (}2\pi \xi (x+t){\bigr )}+a_{-}(\xi )\cos {\bigl (}2\pi \xi (x-t){\bigr )}+{}\\&b_{+}(\xi )\sin {\bigl (}2\pi \xi (x+t){\bigr )}+b_{-}(\xi )\sin \left(2\pi \xi (x-t)\right){\Bigr ]}\end{aligned}}}"> satisfies the wave equation for arbitrary a+, a−, b+, b−. This integral may be interpreted as a continuous linear combination of solutions for the linear equation. Now this resembles the formula for the Fourier synthesis of a function. In fact, this is the real inverse Fourier transform of a± and b± in the variable x. The third step is to examine how to find the specific unknown coefficient functions a± and b± that will lead to y satisfying the boundary conditions. We are interested in the values of these solutions at t = 0. So we will set t = 0. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable x) of both sides and obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\cos(2\pi \xi x)\,dx=a_{+}+a_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\cos(2\pi \xi x)\,dx=a_{+}+a_{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9693cf22844d9148a76946c8052c057eb01223c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.388ex; height:6.009ex;" alt="{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\cos(2\pi \xi x)\,dx=a_{+}+a_{-}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\sin(2\pi \xi x)\,dx=b_{+}+b_{-}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\sin(2\pi \xi x)\,dx=b_{+}+b_{-}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca66228897ef241c9a66452596b1658b8f45afc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.315ex; height:6.009ex;" alt="{\displaystyle 2\int _{-\infty }^{\infty }y(x,0)\sin(2\pi \xi x)\,dx=b_{+}+b_{-}.}"> Similarly, taking the derivative of y with respect to t and then applying the Fourier sine and cosine transformations yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\sin(2\pi \xi x)\,dx=(2\pi \xi )\left(-a_{+}+a_{-}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\sin(2\pi \xi x)\,dx=(2\pi \xi )\left(-a_{+}+a_{-}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d48f65f5ff203f32f4d7070244e84d0fc2853f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.626ex; height:6.343ex;" alt="{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\sin(2\pi \xi x)\,dx=(2\pi \xi )\left(-a_{+}+a_{-}\right)}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\cos(2\pi \xi x)\,dx=(2\pi \xi )\left(b_{+}-b_{-}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\cos(2\pi \xi x)\,dx=(2\pi \xi )\left(b_{+}-b_{-}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe669ddeb80ef4dfc9860a010af298df7e774497" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.642ex; height:6.343ex;" alt="{\displaystyle 2\int _{-\infty }^{\infty }{\frac {\partial y(u,0)}{\partial t}}\cos(2\pi \xi x)\,dx=(2\pi \xi )\left(b_{+}-b_{-}\right).}"> These are four linear equations for the four unknowns a± and b±, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found. In summary, we chose a set of elementary solutions, parametrized by ξ, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter ξ. But this integral was in the form of a Fourier integral. The next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions f and g. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative. The last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions a± and b± in terms of the given boundary conditions f and g. From a higher point of view, Fourier's procedure can be reformulated more conceptually. Since there are two variables, we will use the Fourier transformation in both x and t rather than operate as Fourier did, who only transformed in the spatial variables. Note that ŷ must be considered in the sense of a distribution since y(x, t) is not going to be L1: as a wave, it will persist through time and thus is not a transient phenomenon. But it will be bounded and so its Fourier transform can be defined as a distribution. The operational properties of the Fourier transformation that are relevant to this equation are that it takes differentiation in x to multiplication by i2πξ and differentiation with respect to t to multiplication by i2πf where f is the frequency. Then the wave equation becomes an algebraic equation in ŷ: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ^{2}{\hat {y}}(\xi ,f)=f^{2}{\hat {y}}(\xi ,f).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ^{2}{\hat {y}}(\xi ,f)=f^{2}{\hat {y}}(\xi ,f).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ebb20cf08a4a72cc16821bd05147484b1ff7814" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.116ex; height:3.176ex;" alt="{\displaystyle \xi ^{2}{\hat {y}}(\xi ,f)=f^{2}{\hat {y}}(\xi ,f).}"> This is equivalent to requiring ŷ(ξ, f) = 0 unless ξ = ±f. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously f̂ = δ(ξ ± f) will be solutions. Applying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier. But from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic ξ2 − f2 = 0. We may as well consider the distributions supported on the conic that are given by distributions of one variable on the line ξ = f plus distributions on the line ξ = −f as follows: if Φ is any test function, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iint {\hat {y}}\phi (\xi ,f)\,d\xi \,df=\int s_{+}\phi (\xi ,\xi )\,d\xi +\int s_{-}\phi (\xi ,-\xi )\,d\xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>f</mi> <mo>=</mo> <mo>∫</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>+</mo> <mo>∫</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint {\hat {y}}\phi (\xi ,f)\,d\xi \,df=\int s_{+}\phi (\xi ,\xi )\,d\xi +\int s_{-}\phi (\xi ,-\xi )\,d\xi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc1dc16d94e02f92e90426fe898070fa2bbbfaa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.763ex; height:5.676ex;" alt="{\displaystyle \iint {\hat {y}}\phi (\xi ,f)\,d\xi \,df=\int s_{+}\phi (\xi ,\xi )\,d\xi +\int s_{-}\phi (\xi ,-\xi )\,d\xi ,}"> where s+, and s−, are distributions of one variable. Then Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put Φ(ξ, f) = ei2π(xξ+tf), which is clearly of polynomial growth): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x,0)=\int {\bigl \{}s_{+}(\xi )+s_{-}(\xi ){\bigr \}}e^{i2\pi \xi x+0}\,d\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x,0)=\int {\bigl \{}s_{+}(\xi )+s_{-}(\xi ){\bigr \}}e^{i2\pi \xi x+0}\,d\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a28d85b1667159af9fa50469da1e516005c06f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.651ex; height:5.676ex;" alt="{\displaystyle y(x,0)=\int {\bigl \{}s_{+}(\xi )+s_{-}(\xi ){\bigr \}}e^{i2\pi \xi x+0}\,d\xi }"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y(x,0)}{\partial t}}=\int {\bigl \{}s_{+}(\xi )-s_{-}(\xi ){\bigr \}}i2\pi \xi e^{i2\pi \xi x+0}\,d\xi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y(x,0)}{\partial t}}=\int {\bigl \{}s_{+}(\xi )-s_{-}(\xi ){\bigr \}}i2\pi \xi e^{i2\pi \xi x+0}\,d\xi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874a569545cdfc209e0aa0f08fd813fc3bb75817" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.779ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial y(x,0)}{\partial t}}=\int {\bigl \{}s_{+}(\xi )-s_{-}(\xi ){\bigr \}}i2\pi \xi e^{i2\pi \xi x+0}\,d\xi .}"> Now, as before, applying the one-variable Fourier transformation in the variable x to these functions of x yields two equations in the two unknown distributions s± (which can be taken to be ordinary functions if the boundary conditions are L1 or L2). From a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform. Closed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate. For practical calculations, other methods are often used. The twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well. <div class="mw-heading mw-heading3"><h3 id="Fourier-transform_spectroscopy">Fourier-transform spectroscopy</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=55" title="Edit section: Fourier-transform spectroscopy">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fourier-transform_spectroscopy" title="Fourier-transform spectroscopy">Fourier-transform spectroscopy</a></div> The Fourier transform is also used in <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a> (NMR) and in other kinds of <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>, e.g. infrared (<a href="/wiki/Fourier-transform_infrared_spectroscopy" title="Fourier-transform infrared spectroscopy">FTIR</a>). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> (MRI) and <a href="/wiki/Mass_spectrometry" title="Mass spectrometry">mass spectrometry</a>. <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=56" title="Edit section: Quantum mechanics">edit</a>]</div> The Fourier transform is useful in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of <a href="/wiki/Complementary_variables" class="mw-redirect" title="Complementary variables">complementary variables</a>, connected by the <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>. For example, in one dimension, the spatial variable q of, say, a particle, can only be measured by the quantum mechanical "<a href="/wiki/Position_operator" title="Position operator">position operator</a>" at the cost of losing information about the momentum p of the particle. Therefore, the physical state of the particle can either be described by a function, called "the wave function", of q or by a function of p but not by a function of both variables. The variable p is called the conjugate variable to q. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both p and q simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a p-axis and a q-axis called the <a href="/wiki/Phase_space" title="Phase space">phase space</a>. In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the q-axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis. Nevertheless, choosing the p-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle. Both representations of the wavefunction are related by a Fourier transform, such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (p)=\int dq\,\psi (q)e^{-ipq/h},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (p)=\int dq\,\psi (q)e^{-ipq/h},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584edfbf8d7fb67d88198f57cb04d10cf763ebc4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.268ex; height:5.676ex;" alt="{\displaystyle \phi (p)=\int dq\,\psi (q)e^{-ipq/h},}"> or, equivalently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (q)=\int dp\,\phi (p)e^{ipq/h}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>p</mi> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (q)=\int dp\,\phi (p)e^{ipq/h}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d110dcd432f32d3af4d738b2b569766f1ceb15bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.09ex; height:5.676ex;" alt="{\displaystyle \psi (q)=\int dp\,\phi (p)e^{ipq/h}.}"> Physically realisable states are L2, and so by the <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a>, their Fourier transforms are also L2. (Note that since q is in units of distance and p is in units of momentum, the presence of the Planck constant in the exponent makes the exponent <a href="/wiki/Nondimensionalization" title="Nondimensionalization">dimensionless</a>, as it should be.) Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum. Infinitely many different polarisations are possible, and all are equally valid. Being able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg <a href="#Uncertainty_principle">uncertainty principle</a>. The other use of the Fourier transform in both quantum mechanics and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> is to solve the applicable wave equation. In non-relativistic quantum mechanics, <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a> for a time-varying wave function in one-dimension, not subject to external forces, is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8dfca86d44876c96db0b73927ae2ec97f20f6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.27ex; height:6.009ex;" alt="{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}"> This is the same as the heat equation except for the presence of the imaginary unit i. Fourier methods can be used to solve this equation. In the presence of a potential, given by the potential energy function V(x), the equation becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010bed8fcaa500618b36fbabd66bd57fced5ffe1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:44.562ex; height:6.009ex;" alt="{\displaystyle -{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t)=i{\frac {h}{2\pi }}{\frac {\partial }{\partial t}}\psi (x,t).}"> The "elementary solutions", as we referred to them above, are the so-called "stationary states" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of ψ given its values for t = 0. Neither of these approaches is of much practical use in quantum mechanics. Boundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important. In relativistic quantum mechanics, Schrödinger's equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered. A simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+1\right)\psi (x,t)={\frac {\partial ^{2}}{\partial t^{2}}}\psi (x,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+1\right)\psi (x,t)={\frac {\partial ^{2}}{\partial t^{2}}}\psi (x,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356c1d560b7872590dfcf4fbacf885e84ad341a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.194ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+1\right)\psi (x,t)={\frac {\partial ^{2}}{\partial t^{2}}}\psi (x,t).}"> This is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods). This is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as "second quantization". Fourier methods have been adapted to also deal with non-trivial interactions. Finally, the <a href="/wiki/Quantum_harmonic_oscillator#Ladder_operator_method" title="Quantum harmonic oscillator">number operator</a> of the <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> can be interpreted, for example via the <a href="/wiki/Mehler_kernel#Physics_version" title="Mehler kernel">Mehler kernel</a>, as the <a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">generator</a> of the <a href="#Eigenfunctions">Fourier transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">.<a href="#cite_note-auto-32">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Signal_processing">Signal processing</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=57" title="Edit section: Signal processing">edit</a>]</div> The Fourier transform is used for the spectral analysis of time-series. The subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself. Even if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time. The Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function. The autocorrelation function R of a function f is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{f}(\tau )=\lim _{T\rightarrow \infty }{\frac {1}{2T}}\int _{-T}^{T}f(t)f(t+\tau )\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{f}(\tau )=\lim _{T\rightarrow \infty }{\frac {1}{2T}}\int _{-T}^{T}f(t)f(t+\tau )\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5fc92d80f09707e5ee1c46f241cf9fbb50a547f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.197ex; height:6.343ex;" alt="{\displaystyle R_{f}(\tau )=\lim _{T\rightarrow \infty }{\frac {1}{2T}}\int _{-T}^{T}f(t)f(t+\tau )\,dt.}"> This function is a function of the time-lag τ elapsing between the values of f to be correlated. For most functions f that occur in practice, R is a bounded even function of the time-lag τ and for typical noisy signals it turns out to be uniformly continuous with a maximum at τ = 0. The autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values of f separated by a time lag. This is a way of searching for the correlation of f with its own past. It is useful even for other statistical tasks besides the analysis of signals. For example, if f(t) represents the temperature at time t, one expects a strong correlation with the temperature at a time lag of 24 hours. It possesses a Fourier transform, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{f}(\xi )=\int _{-\infty }^{\infty }R_{f}(\tau )e^{-i2\pi \xi \tau }\,d\tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>τ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{f}(\xi )=\int _{-\infty }^{\infty }R_{f}(\tau )e^{-i2\pi \xi \tau }\,d\tau .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5585cc5177d458dca7abfa5e9b8b7fb515e9b1cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.267ex; height:6.009ex;" alt="{\displaystyle P_{f}(\xi )=\int _{-\infty }^{\infty }R_{f}(\tau )e^{-i2\pi \xi \tau }\,d\tau .}"> This Fourier transform is called the <a href="/wiki/Spectral_density#Power_spectral_density" title="Spectral density">power spectral density</a> function of f. (Unless all periodic components are first filtered out from f, this integral will diverge, but it is easy to filter out such periodicities.) The power spectrum, as indicated by this density function P, measures the amount of variance contributed to the data by the frequency ξ. In electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal. This process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series (<a href="/wiki/ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a>). Knowledge of which frequencies are "important" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses. It can also be useful for the scientific analysis of the phenomena responsible for producing the data. The power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out. Spectral analysis is carried out for visual signals as well. The power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool. <div class="mw-heading mw-heading2"><h2 id="Other_notations">Other notations</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=58" title="Edit section: Other notations">edit</a>]</div> Other common notations for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c12179a82497132d68a556a537ef57b46cbbf83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.538ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )}"> include: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}(\xi ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}{\bigl (}f(t){\bigr )},\ {\mathcal {F}}{\bigl \{}f(t){\bigr \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(\xi ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}{\bigl (}f(t){\bigr )},\ {\mathcal {F}}{\bigl \{}f(t){\bigr \}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c626f6f979558bd9084c00012956920952beb137" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:65.417ex; height:3.343ex;" alt="{\displaystyle {\tilde {f}}(\xi ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}{\bigl (}f(t){\bigr )},\ {\mathcal {F}}{\bigl \{}f(t){\bigr \}}.}"> In the sciences and engineering it is also common to make substitutions like these: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \rightarrow f,\quad x\rightarrow t,\quad f\rightarrow x,\quad {\hat {f}}\rightarrow X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo stretchy="false">→</mo> <mi>f</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \rightarrow f,\quad x\rightarrow t,\quad f\rightarrow x,\quad {\hat {f}}\rightarrow X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a7421704e52e9e335ec8166eb177a7210d6cc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.938ex; height:3.176ex;" alt="{\displaystyle \xi \rightarrow f,\quad x\rightarrow t,\quad f\rightarrow x,\quad {\hat {f}}\rightarrow X.}"> So the transform pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb38d71da2c23f03006b608a37cae2fa26e6d41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.434ex; height:4.176ex;" alt="{\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )}"> can become <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a3b88136ceceebe84bc5f9486003c9a41a8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.524ex; height:4.176ex;" alt="{\displaystyle x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)}"> A disadvantage of the capital letter notation is when expressing a transform such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f\cdot g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo>⋅</mo> <mi>g</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f\cdot g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7958fc7cccb80dbeee2f9b9d7f01622d38f309e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.074ex; height:3.343ex;" alt="{\displaystyle {\widehat {f\cdot g}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f'}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f'}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49770579012f63235d959305a6c5fc2a82dce1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.01ex; width:2.981ex; height:3.509ex;" alt="{\displaystyle {\widehat {f'}},}"> which become the more awkward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{f\cdot g\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>⋅</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f\cdot g\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06208068776598b397c039b385dd541c75cafcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.325ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{f\cdot g\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{f'\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f'\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5974e5904c4447aac2f04376ff607eedb8aa4174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.904ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\{f'\}.}"> In some contexts such as particle physics, the same symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> may be used for both for a function as well as it Fourier transform, with the two only distinguished by their <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> I.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k_{1}+k_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(k_{1}+k_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67bcce27ed30ce33b34a1968c804b307d2f18e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.459ex; height:2.843ex;" alt="{\displaystyle f(k_{1}+k_{2})}"> would refer to the Fourier transform because of the momentum argument, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0}+\pi {\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{0}+\pi {\vec {r}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a3241124d1886d982128c6d45a9b4454d807eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.867ex; height:2.843ex;" alt="{\displaystyle f(x_{0}+\pi {\vec {r}})}"> would refer to the original function because of the positional argument. Although tildes may be used as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6cb99679a4b79cb5ca3c242811bd91220c91f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}}"> to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a> form, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {dk}}={\frac {dk}{(2\pi )^{3}2\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>k</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>k</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>2</mn> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {dk}}={\frac {dk}{(2\pi )^{3}2\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752aee376c172cd8f45590c7c9e779a293cb05e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.328ex; height:6.176ex;" alt="{\displaystyle {\tilde {dk}}={\frac {dk}{(2\pi )^{3}2\omega }}}">, so care must be taken. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"> often denotes the <a href="/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. The interpretation of the complex function f̂(ξ) may be aided by expressing it in <a href="/wiki/Polar_coordinate" class="mw-redirect" title="Polar coordinate">polar coordinate</a> form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0055a14c74932c1b4d8cea21eeebe53f832b5efe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.185ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}}"> in terms of the two real functions A(ξ) and φ(ξ) where: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\xi )=\left|{\hat {f}}(\xi )\right|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(\xi )=\left|{\hat {f}}(\xi )\right|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fd7058b1090db104e3220ec98e4f480b7edece" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.547ex; height:3.843ex;" alt="{\displaystyle A(\xi )=\left|{\hat {f}}(\xi )\right|,}"> is the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\xi )=\arg \left({\hat {f}}(\xi )\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (\xi )=\arg \left({\hat {f}}(\xi )\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d498b50a32866a93cb61d44b63b2e6a81fbcd75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.655ex; height:4.843ex;" alt="{\displaystyle \varphi (\xi )=\arg \left({\hat {f}}(\xi )\right),}"> is the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> (see <a href="/wiki/Arg_(mathematics)" class="mw-redirect" title="Arg (mathematics)">arg function</a>). Then the inverse transform can be written: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i{\bigl (}2\pi \xi x+\varphi (\xi ){\bigr )}}\,d\xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i{\bigl (}2\pi \xi x+\varphi (\xi ){\bigr )}}\,d\xi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f9516925ac72f194e912b0f603281afc0e2a0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.432ex; height:6.009ex;" alt="{\displaystyle f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i{\bigl (}2\pi \xi x+\varphi (\xi ){\bigr )}}\,d\xi ,}"> which is a recombination of all the frequency components of f(x). Each component is a complex <a href="/wiki/Sinusoid" class="mw-redirect" title="Sinusoid">sinusoid</a> of the form e2πixξ whose amplitude is A(ξ) and whose initial <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase angle</a> (at x = 0) is φ(ξ). The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted F and F(f) is used to denote the Fourier transform of the function f. This mapping is linear, which means that F can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function f) can be used to write F f instead of F(f). Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as F f(ξ) or as (F f)(ξ). Notice that in the former case, it is implicitly understood that F is applied first to f and then the resulting function is evaluated at ξ, not the other way around. In mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted f(x). This means that a notation like F(f(x)) formally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}{\bigl (}\operatorname {rect} (x){\bigr )}=\operatorname {sinc} (\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}{\bigl (}\operatorname {rect} (x){\bigr )}=\operatorname {sinc} (\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e835a41babdadd8485f4841769c216edff7bbef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.289ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}{\bigl (}\operatorname {rect} (x){\bigr )}=\operatorname {sinc} (\xi )}"> is sometimes used to express that the Fourier transform of a <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> is a <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>, or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}{\bigl (}f(x+x_{0}){\bigr )}={\mathcal {F}}{\bigl (}f(x){\bigr )}\,e^{i2\pi x_{0}\xi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ξ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}{\bigl (}f(x+x_{0}){\bigr )}={\mathcal {F}}{\bigl (}f(x){\bigr )}\,e^{i2\pi x_{0}\xi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bead7983a3d2aa5bcf5fefd721a1f82299df7de0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.805ex; height:3.343ex;" alt="{\displaystyle {\mathcal {F}}{\bigl (}f(x+x_{0}){\bigr )}={\mathcal {F}}{\bigl (}f(x){\bigr )}\,e^{i2\pi x_{0}\xi }}"> is used to express the shift property of the Fourier transform. Notice, that the last example is only correct under the assumption that the transformed function is a function of x, not of x0. As discussed above, the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of a random variable is the same as the <a href="#Fourier–Stieltjes_transform">Fourier–Stieltjes transform</a> of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(e^{it\cdot X}\right)=\int e^{it\cdot x}\,d\mu _{X}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mo>⋅</mo> <mi>X</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left(e^{it\cdot X}\right)=\int e^{it\cdot x}\,d\mu _{X}(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2910084ada38e0029db77258571d6f99323f33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.603ex; height:5.676ex;" alt="{\displaystyle E\left(e^{it\cdot X}\right)=\int e^{it\cdot x}\,d\mu _{X}(x).}"> As in the case of the "non-unitary angular frequency" convention above, the factor of 2π appears in neither the normalizing constant nor the exponent. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponent. <div class="mw-heading mw-heading2"><h2 id="Computation_methods">Computation methods</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=59" title="Edit section: Computation methods">edit</a>]</div> The appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function. In this section we consider both functions of a continuous variable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"> and functions of a discrete variable (i.e. ordered pairs of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> values). For discrete-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }">). When the sinusoids are harmonically-related (i.e. when the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-values are spaced at integer multiples of an interval), the transform is called <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT). <div class="mw-heading mw-heading3"><h3 id="Discrete_Fourier_transforms_and_fast_Fourier_transforms">Discrete Fourier transforms and fast Fourier transforms</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=60" title="Edit section: Discrete Fourier transforms and fast Fourier transforms">edit</a>]</div> Sampling the DTFT at equally-spaced values of frequency is the most common modern method of computation. Efficient procedures, depending on the frequency resolution needed, are described at <a href="/wiki/Discrete-time_Fourier_transform#Sampling_the_DTFT" title="Discrete-time Fourier transform">Discrete-time Fourier transform § Sampling the DTFT</a>. The <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT), used there, is usually computed by a <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) algorithm. <div class="mw-heading mw-heading3"><h3 id="Analytic_integration_of_closed-form_functions">Analytic integration of closed-form functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=61" title="Edit section: Analytic integration of closed-form functions">edit</a>]</div> Tables of <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form</a> Fourier transforms, such as <a href="#Square-integrable_functions,_one-dimensional">§ Square-integrable functions, one-dimensional</a> and <a href="/wiki/Discrete-time_Fourier_transform#Table_of_discrete-time_Fourier_transforms" title="Discrete-time Fourier transform">§ Table of discrete-time Fourier transforms</a>, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }">).<a href="#cite_note-Zwillinger-2014-62">[56]</a> When mathematically possible, this provides a transform for a continuum of frequency values. Many computer algebra systems such as <a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a> and <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> that are capable of <a href="/wiki/Symbolic_integration" title="Symbolic integration">symbolic integration</a> are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform of cos(6πt) e−πt2 one might enter the command <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf</code> into <a href="/wiki/Wolfram_Alpha" class="mw-redirect" title="Wolfram Alpha">Wolfram Alpha</a>.<a href="#cite_note-63">[note 7]</a> <div class="mw-heading mw-heading3"><h3 id="Numerical_integration_of_closed-form_continuous_functions">Numerical integration of closed-form continuous functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=62" title="Edit section: Numerical integration of closed-form continuous functions">edit</a>]</div> Discrete sampling of the Fourier transform can also be done by <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a> of the definition at each value of frequency for which transform is desired.<a href="#cite_note-64">[57]</a><a href="#cite_note-65">[58]</a><a href="#cite_note-66">[59]</a> The numerical integration approach works on a much broader class of functions than the analytic approach. <div class="mw-heading mw-heading3"><h3 id="Numerical_integration_of_a_series_of_ordered_pairs">Numerical integration of a series of ordered pairs</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=63" title="Edit section: Numerical integration of a series of ordered pairs">edit</a>]</div> If the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.<a href="#cite_note-67">[60]</a> The DTFT is a common subcase of this more general situation. <div class="mw-heading mw-heading2"><h2 id="Tables_of_important_Fourier_transforms">Tables of important Fourier transforms</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=64" title="Edit section: Tables of important Fourier transforms">edit</a>]</div> The following tables record some closed-form Fourier transforms. For functions f(x) and g(x) denote their Fourier transforms by f̂ and ĝ. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse. <div class="mw-heading mw-heading3"><h3 id="Functional_relationships,_one-dimensional">Functional relationships, one-dimensional</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=65" title="Edit section: Functional relationships, one-dimensional">edit</a>]</div> The Fourier transforms in this table may be found in <a href="#CITEREFErdélyi1954">Erdélyi (1954)</a> or <a href="#CITEREFKammler2000">Kammler (2000</a>, appendix). <table class="wikitable"> <tbody><tr> <th></th> <th>Function</th> <th>Fourier transform unitary, ordinary frequency</th> <th>Fourier transform unitary, angular frequency</th> <th>Fourier transform non-unitary, angular frequency</th> <th>Remarks </th></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\widehat {f}}(\xi )\triangleq {\widehat {f_{1}}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\widehat {f}}(\xi )\triangleq {\widehat {f_{1}}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d39935a56ec3b5a6e8933dbba043189874ea941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.987ex; margin-bottom: -0.185ex; width:22.628ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\widehat {f}}(\xi )\triangleq {\widehat {f_{1}}}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{2}}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{2}}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/031d4f07e364b1bb4b429ce2cfcb669121808faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.812ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{2}}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{3}}}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{3}}}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900caef6aeeef7d97ca479d8fd6602fd8e0c030b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.987ex; margin-bottom: -0.185ex; width:21.158ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\widehat {f}}(\omega )\triangleq {\widehat {f_{3}}}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td>Definitions </td></tr> <tr> <td>101 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,f(x)+b\,g(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,f(x)+b\,g(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9424bce4a1d7a6bbe0e802e8627f7ddd13256381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.902ex; height:2.843ex;" alt="{\displaystyle a\,f(x)+b\,g(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\widehat {f}}(\xi )+b\,{\widehat {g}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\widehat {f}}(\xi )+b\,{\widehat {g}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc08b3a12d22b1f1000ce6db9cc4aa8556de75e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.91ex; height:3.509ex;" alt="{\displaystyle a\,{\widehat {f}}(\xi )+b\,{\widehat {g}}(\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a528c84510360232e791d11cd97ea60b3db3103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.742ex; height:3.509ex;" alt="{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a528c84510360232e791d11cd97ea60b3db3103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.742ex; height:3.509ex;" alt="{\displaystyle a\,{\widehat {f}}(\omega )+b\,{\widehat {g}}(\omega )\,}"> </td> <td>Linearity </td></tr> <tr> <td>102 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x-a)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x-a)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7715e034cf2f100e5e18cd179d13f39414c8de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.875ex; height:2.843ex;" alt="{\displaystyle f(x-a)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i2\pi \xi a}{\widehat {f}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>a</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i2\pi \xi a}{\widehat {f}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fb0a4bdcf8855cad006777d27a2e0461efb681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.521ex; height:3.509ex;" alt="{\displaystyle e^{-i2\pi \xi a}{\widehat {f}}(\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>ω</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bdd39d1518ca5fcd9a09de93138e80a49199ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.467ex; height:3.509ex;" alt="{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>ω</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bdd39d1518ca5fcd9a09de93138e80a49199ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.467ex; height:3.509ex;" alt="{\displaystyle e^{-ia\omega }{\widehat {f}}(\omega )\,}"> </td> <td>Shift in time domain </td></tr> <tr> <td>103 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)e^{iax}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)e^{iax}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5a5c5939d4d7a4be7e34167d6b30ea2a760020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.498ex; height:3.176ex;" alt="{\displaystyle f(x)e^{iax}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/922505ea0a8bf929cd037aed8c11d651d11f010e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.522ex; height:4.843ex;" alt="{\displaystyle {\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\omega -a)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\omega -a)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8631bc96e99bfb42de530ad9a8dfd62a38ef00dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.483ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\omega -a)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\omega -a)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\omega -a)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8631bc96e99bfb42de530ad9a8dfd62a38ef00dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.483ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\omega -a)\,}"> </td> <td>Shift in frequency domain, dual of 102 </td></tr> <tr> <td>104 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(ax)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111bb5a0bfcace9d4bda71be32da87798f5c1cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.034ex; height:2.843ex;" alt="{\displaystyle f(ax)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ae2be1a25afc9018e30da0ae4c139fb568cf9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.392ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\xi }{a}}\right)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c28ff9bc66d8c50205881392d1e34c0ee9a532e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.962ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c28ff9bc66d8c50205881392d1e34c0ee9a532e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.962ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}{\widehat {f}}\left({\frac {\omega }{a}}\right)\,}"> </td> <td>Scaling in the time domain. If |a| is large, then f(ax) is concentrated around 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be2f1a48f9dcd01e0322f2cc8dbf6f79b31954e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.89ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}"> spreads out and flattens. </td></tr> <tr> <td>105 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f_{n}}}(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f_{n}}}(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7dd1477c790a184ab8ec9ca5da4cddbf33ebf08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.884ex; height:3.509ex;" alt="{\displaystyle {\widehat {f_{n}}}(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f_{1}}}(x)\ {\stackrel {{\mathcal {F}}_{1}}{\longleftrightarrow }}\ f(-\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟷</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f_{1}}}(x)\ {\stackrel {{\mathcal {F}}_{1}}{\longleftrightarrow }}\ f(-\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0704f808ad7d6cebe128f9a172a9754423a5ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:17.267ex; height:4.176ex;" alt="{\displaystyle {\widehat {f_{1}}}(x)\ {\stackrel {{\mathcal {F}}_{1}}{\longleftrightarrow }}\ f(-\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f_{2}}}(x)\ {\stackrel {{\mathcal {F}}_{2}}{\longleftrightarrow }}\ f(-\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟷</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f_{2}}}(x)\ {\stackrel {{\mathcal {F}}_{2}}{\longleftrightarrow }}\ f(-\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74ea3e8e5ba2e98f61156c02ee89462a69ab9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:17.682ex; height:4.176ex;" alt="{\displaystyle {\widehat {f_{2}}}(x)\ {\stackrel {{\mathcal {F}}_{2}}{\longleftrightarrow }}\ f(-\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f_{3}}}(x)\ {\stackrel {{\mathcal {F}}_{3}}{\longleftrightarrow }}\ 2\pi f(-\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟷</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f_{3}}}(x)\ {\stackrel {{\mathcal {F}}_{3}}{\longleftrightarrow }}\ 2\pi f(-\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b16c12ff2fdbb745d4ffa06e486754c0875907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:20.177ex; height:4.176ex;" alt="{\displaystyle {\widehat {f_{3}}}(x)\ {\stackrel {{\mathcal {F}}_{3}}{\longleftrightarrow }}\ 2\pi f(-\omega )\,}"> </td> <td>The same transform is applied twice, but x replaces the frequency variable (ξ or ω) after the first transform. </td></tr> <tr> <td>106 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0a54d8e55c6c2bf3fbdc50891df151c2dfcce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.077ex; height:5.843ex;" alt="{\displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i2\pi \xi )^{n}{\widehat {f}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i2\pi \xi )^{n}{\widehat {f}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014d73aef42d5b6f17d65cb17981088bdc40f113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.352ex; height:3.509ex;" alt="{\displaystyle (i2\pi \xi )^{n}{\widehat {f}}(\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7067214c7c442df051c39f7a3c80185ddb0dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.689ex; height:3.509ex;" alt="{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7067214c7c442df051c39f7a3c80185ddb0dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.689ex; height:3.509ex;" alt="{\displaystyle (i\omega )^{n}{\widehat {f}}(\omega )\,}"> </td> <td>nth-order derivative. As f is a <a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz function</a> </td></tr> <tr> <td>106.5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{x}f(\tau )d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{x}f(\tau )d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f68020f554eeb49cc58a188ae622a54bde5102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.541ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{x}f(\tau )d\tau }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\xi )}{i2\pi \xi }}+C\,\delta (\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\xi )}{i2\pi \xi }}+C\,\delta (\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3103b12fd1929645ba4eca63efab3118e696a4ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.328ex; height:6.676ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\xi )}{i2\pi \xi }}+C\,\delta (\xi )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+{\sqrt {2\pi }}C\delta (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mi>C</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+{\sqrt {2\pi }}C\delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a68e813680cf13afbf82eaa08af679b147886b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.203ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+{\sqrt {2\pi }}C\delta (\omega )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+2\pi C\delta (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>C</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+2\pi C\delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47d25eacccc83e6fa7a8acd32a0f5fff6abba25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.267ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega )}{i\omega }}+2\pi C\delta (\omega )}"> </td> <td>Integration.<a href="#cite_note-68">[61]</a> Note: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is the average (<a href="/wiki/DC_component" class="mw-redirect" title="DC component">DC</a>) value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }(f(x)-C)\,dx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }(f(x)-C)\,dx=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8091d15a283d7c8a2961ebba9ab15ad576006f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.474ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }(f(x)-C)\,dx=0}"> </td></tr> <tr> <td>107 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}f(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3b71dbeac2bea1f8a23a0a53bed91f11141bdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.353ex; height:2.843ex;" alt="{\displaystyle x^{n}f(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\widehat {f}}(\xi )}{d\xi ^{n}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\widehat {f}}(\xi )}{d\xi ^{n}}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f8f215cadc8e2fed661eb973991038dd29f99d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.24ex; height:6.843ex;" alt="{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\widehat {f}}(\xi )}{d\xi ^{n}}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2dbcf7d5096502b5650ca453321402e2db0c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.32ex; height:6.343ex;" alt="{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2dbcf7d5096502b5650ca453321402e2db0c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.32ex; height:6.343ex;" alt="{\displaystyle i^{n}{\frac {d^{n}{\widehat {f}}(\omega )}{d\omega ^{n}}}}"> </td> <td>This is the dual of 106 </td></tr> <tr> <td>108 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434a18852bf258f44f2e3b4f4c687b33e37102f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.925ex; height:2.843ex;" alt="{\displaystyle (f*g)(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi ){\widehat {g}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi ){\widehat {g}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6698c9ffc5ebba936125c3a09e86a3b256084f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.068ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi ){\widehat {g}}(\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}\ {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\ {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d1fd0a5f01c2e7a28cabe33eba88b45404fdb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.911ex; height:3.509ex;" alt="{\displaystyle {\sqrt {2\pi }}\ {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/febc07d3226c12030e627d9ff92b703a206b6b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\omega ){\widehat {g}}(\omega )\,}"> </td> <td>The notation f ∗ g denotes the <a href="/wiki/Convolution" title="Convolution">convolution</a> of f and g — this rule is the <a href="/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a> </td></tr> <tr> <td>109 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)g(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)g(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0793a5d394df8f22dc913ab9fd856cc869f20440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.06ex; height:2.843ex;" alt="{\displaystyle f(x)g(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\widehat {f}}*{\widehat {g}}\right)(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\widehat {f}}*{\widehat {g}}\right)(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974a30f4cda0b28fe407843391b0f18d8219b8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.586ex; height:4.843ex;" alt="{\displaystyle \left({\widehat {f}}*{\widehat {g}}\right)(\xi )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161655edf2131f9c32eb58002a066f675ac168fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.656ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c99f1091da8efc3f07f5647dc699b77abbe5f5de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.72ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}\left({\widehat {f}}*{\widehat {g}}\right)(\omega )\,}"> </td> <td>This is the dual of 108 </td></tr> <tr> <td>110 </td> <td>For f(x) purely real </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\xi )={\overline {{\widehat {f}}(\xi )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\xi )={\overline {{\widehat {f}}(\xi )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eaa03948fc6041a0cf979e2eb5a219bb5d457b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.629ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\xi )={\overline {{\widehat {f}}(\xi )}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2f26c94c42eca741887346d75c1a4f4173ca22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.461ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2f26c94c42eca741887346d75c1a4f4173ca22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.461ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\omega )={\overline {{\widehat {f}}(\omega )}}\,}"> </td> <td>Hermitian symmetry. z indicates the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>. </td></tr> <tr> <td>113 </td> <td>For f(x) purely imaginary </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\xi )=-{\overline {{\widehat {f}}(\xi )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\xi )=-{\overline {{\widehat {f}}(\xi )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee0b2e7dd5f52f7d8b3a9fc724d83bce1cf107d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.438ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\xi )=-{\overline {{\widehat {f}}(\xi )}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f495f0543a14d0594d2a8c7d1426b92c1154a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.269ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f495f0543a14d0594d2a8c7d1426b92c1154a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.269ex; height:4.176ex;" alt="{\displaystyle {\widehat {f}}(-\omega )=-{\overline {{\widehat {f}}(\omega )}}\,}"> </td> <td>z indicates the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>. </td></tr> <tr> <td>114 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f(x)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24103646428feba4311115a4dd47e947f44358f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.533ex; height:3.676ex;" alt="{\displaystyle {\overline {f(x)}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\widehat {f}}(-\xi )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\widehat {f}}(-\xi )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfe1d544e95803de8a04e5dc9c9a06817e3975a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.533ex; height:4.176ex;" alt="{\displaystyle {\overline {{\widehat {f}}(-\xi )}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998a8cce7bb5d33c45ce5450c6f4f0cf90ae2cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.949ex; height:4.176ex;" alt="{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998a8cce7bb5d33c45ce5450c6f4f0cf90ae2cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.949ex; height:4.176ex;" alt="{\displaystyle {\overline {{\widehat {f}}(-\omega )}}}"></td> <td><a href="/wiki/Complex_conjugate" title="Complex conjugate">Complex conjugation</a>, generalization of 110 and 113 </td></tr> <tr> <td>115 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\cos(ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\cos(ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c68500e07727bcc43d62d70726c172c21e7f49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.285ex; height:2.843ex;" alt="{\displaystyle f(x)\cos(ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e00b2f8192bdcc70471d24d9731d62bfb5d4069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.485ex; height:7.509ex;" alt="{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07055b374f928c482c9fa4f4ba3ff056a3e5ff58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.256ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47504104332ce0ad3c43b793676078a7093b986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.869ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega -a)+{\widehat {f}}(\omega +a)}{2}}}"> </td> <td>This follows from rules 101 and 103 using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32bc4f95d1c7d0df1421d0cd3c842b040302d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.566ex; height:5.676ex;" alt="{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}"> </td></tr> <tr> <td>116 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\sin(ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\sin(ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0ac7705e63a91a332ff572ba2d821f6a390fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.029ex; height:2.843ex;" alt="{\displaystyle f(x)\sin(ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eee6cb36ceb490addc3240c7516bc2db1765297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.485ex; height:7.509ex;" alt="{\displaystyle {\frac {{\widehat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\widehat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed33230bcd180d4ce535de5086d16b7cc4d4395e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.869ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed33230bcd180d4ce535de5086d16b7cc4d4395e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.869ex; height:6.176ex;" alt="{\displaystyle {\frac {{\widehat {f}}(\omega -a)-{\widehat {f}}(\omega +a)}{2i}}}"> </td> <td>This follows from 101 and 103 using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2aebcf3f2435da6664e2bd570be8cb5e65373c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.311ex; height:5.676ex;" alt="{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Square-integrable_functions,_one-dimensional">Square-integrable functions, one-dimensional</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=66" title="Edit section: Square-integrable functions, one-dimensional">edit</a>]</div> The Fourier transforms in this table may be found in <a href="#CITEREFCampbellFoster1948">Campbell & Foster (1948)</a>, <a href="#CITEREFErdélyi1954">Erdélyi (1954)</a>, or <a href="#CITEREFKammler2000">Kammler (2000</a>, appendix). <table class="wikitable"> <tbody><tr> <th></th> <th>Function</th> <th>Fourier transform unitary, ordinary frequency</th> <th>Fourier transform unitary, angular frequency</th> <th>Fourier transform non-unitary, angular frequency</th> <th>Remarks </th></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969a95df9806ad1eb9f1c854afaba1f947d81f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:22.628ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ddf1bb65cf4f6667e4c97dd701404f26308cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.812ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10016c4aacf29d824cab8af653e0e21157785fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.158ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td>Definitions </td></tr> <tr> <td> 201 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rect} (ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} (ax)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5900b8bd21aea1c79721b6c64b5c15d64a3fe7d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.637ex; height:2.843ex;" alt="{\displaystyle \operatorname {rect} (ax)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\xi }{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>sinc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\xi }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21396b9d9506b857fe48319c7f4df02b887ad87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.509ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\xi }{a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>sinc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90fef38dc70790cf5eea41b8904652fb21a0d6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.549ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>sinc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a8502c5b8cd577814a0e2b11762b135e814a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.358ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td>The <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular pulse</a> and the normalized <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a>, here defined as sinc(x) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠sin(πx)/πx⁠ </td></tr> <tr> <td>202 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (ax)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c9929c0b369303061aa6e2b09861ba6e95d6d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.644ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (ax)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\xi }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0335a2fa05d837219325879919e7840b4b0b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.889ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\xi }{a}}\right)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2bfe98add459fd340dfec05688ff1816245ed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.542ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ac2580dcf156a90cbde09aeabcc98ffad32daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.351ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td>Dual of rule 201. The <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> is an ideal <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>, and the <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a> is the <a href="/wiki/Anticausal_system" title="Anticausal system">non-causal</a> impulse response of such a filter. The <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a> is defined here as sinc(x) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠sin(πx)/πx⁠ </td></tr> <tr> <td>203 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} ^{2}(ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} ^{2}(ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3eeb048450332ce0e09cad3b29ed1b61aee0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.311ex; height:3.176ex;" alt="{\displaystyle \operatorname {sinc} ^{2}(ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\xi }{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>tri</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\xi }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/050bfec21019bfff8c558722f1015bb328386f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.084ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\xi }{a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>tri</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4580555650f0afa976e11a1c79aa6ce4040dea4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.124ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>tri</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792f5aab14288abbbe689c3274d076310e5765f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.933ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}"> </td> <td>The function tri(x) is the <a href="/wiki/Triangular_function" title="Triangular function">triangular function</a> </td></tr> <tr> <td>204 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tri} (ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tri</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tri} (ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdee9659d4f8987bfd5e15058b5a2c84e4b76e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle \operatorname {tri} (ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22aa0aa78cd8d4fde331001c8e259af045663a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.95ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b17f28846951e5996c3f4a6fde32cab790d36c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.603ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/549028ce18204042505ef79553031bd1fa34d346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.412ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\,\operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}"> </td> <td>Dual of rule 203. </td></tr> <tr> <td>205 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-ax}u(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-ax}u(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ea3ebffabbae1fa3dd9d273a703ba5370a8fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.26ex; height:3.009ex;" alt="{\displaystyle e^{-ax}u(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a+i2\pi \xi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+i2\pi \xi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b04f51f9bfbddebbafead47309bf61b834ed19e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.233ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{a+i2\pi \xi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e3cb590825e1f490e4ef00ac0f4788ad8be458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.394ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a+i\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+i\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1942220350b002e38eee6326512bb2479ebc0098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.155ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{a+i\omega }}}"> </td> <td>The function u(x) is the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside unit step function</a> and a > 0. </td></tr> <tr> <td>206 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\alpha x^{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha x^{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b752d2bcf4b8a8ee01e78ae77a2ad24d896ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.805ex; height:3.009ex;" alt="{\displaystyle e^{-\alpha x^{2}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>α</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b95277378a6e90ccb0a82a26f5e3a791ab96180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.66ex; height:6.509ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>α</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721f0753fe72636915afb21102a03d737420b044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.87ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>α</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096ecb69bc8c8137c17376e365a8941228fb97bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.096ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\,e^{-{\frac {\omega ^{2}}{4\alpha }}}}"> </td> <td>This shows that, for the unitary Fourier transforms, the <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian function</a> e−αx2 is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α) > 0. </td></tr> <tr> <td>208 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-a|x|}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-a|x|}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da7129dd682ad860a73fd3145aab20e001fadf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.706ex; height:2.843ex;" alt="{\displaystyle e^{-a|x|}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40a975d764eb9941a9f5f4858c9b09bb74e385b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.599ex; height:5.843ex;" alt="{\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {a}{a^{2}+\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {a}{a^{2}+\omega ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d18739930da1c504f6b5302ab794f8f5f7b02b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.34ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {a}{a^{2}+\omega ^{2}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160b3490d077cf0ce5afbddf809f01a0e2cf39ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.461ex; height:5.676ex;" alt="{\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}"> </td> <td>For Re(a) > 0. That is, the Fourier transform of a <a href="/wiki/Laplace_distribution" title="Laplace distribution">two-sided decaying exponential function</a> is a <a href="/wiki/Lorentzian_function" class="mw-redirect" title="Lorentzian function">Lorentzian function</a>. </td></tr> <tr> <td>209 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sech} (ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sech</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sech} (ax)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f2e18489a8cf729be1acc54e94e00280cb4666d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.029ex; height:2.843ex;" alt="{\displaystyle \operatorname {sech} (ax)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </mrow> <mi>sech</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab639d17bcb51b9f42fd84e757e4e93777159bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.505ex; height:6.343ex;" alt="{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mi>sech</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7c391502f9cfdac6976a5dc84a32c7cbc1585b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.668ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </mrow> <mi>sech</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0492081bed48b09186bf1fc88b22b3e88cf9ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.279ex; height:4.843ex;" alt="{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}"> </td> <td><a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">Hyperbolic secant</a> is its own Fourier transform </td></tr> <tr> <td>210 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a1fe6cca47234d10ad440541ea2ec46b6029fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.407ex; height:4.509ex;" alt="{\displaystyle e^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> </msup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2ba5272eb233049cb545d0772c9430f7fa88d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.282ex; height:6.843ex;" alt="{\displaystyle {\frac {{\sqrt {2\pi }}(-i)^{n}}{a}}e^{-{\frac {2\pi ^{2}\xi ^{2}}{a^{2}}}}H_{n}\left({\frac {2\pi \xi }{a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42306c2bf9928bc4f9a0b6045444936e3d1dd9fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.673ex; height:6.009ex;" alt="{\displaystyle {\frac {(-i)^{n}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </msup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4aa7015f0a2dbc22d57729ea8b69e0b3ad369f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.103ex; height:6.009ex;" alt="{\displaystyle {\frac {(-i)^{n}{\sqrt {2\pi }}}{a}}e^{-{\frac {\omega ^{2}}{2a^{2}}}}H_{n}\left({\frac {\omega }{a}}\right)}"> </td> <td>Hn is the nth-order <a href="/wiki/Hermite_polynomial" class="mw-redirect" title="Hermite polynomial">Hermite polynomial</a>. If a = 1 then the Gauss–Hermite functions are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the Fourier transform operator. For a derivation, see <a href="/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform" title="Hermite polynomials">Hermite polynomial</a>. The formula reduces to 206 for n = 0. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Distributions,_one-dimensional">Distributions, one-dimensional</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=67" title="Edit section: Distributions, one-dimensional">edit</a>]</div> The Fourier transforms in this table may be found in <a href="#CITEREFErdélyi1954">Erdélyi (1954)</a> or <a href="#CITEREFKammler2000">Kammler (2000</a>, appendix). <table class="wikitable"> <tbody><tr> <th></th> <th>Function</th> <th>Fourier transform unitary, ordinary frequency</th> <th>Fourier transform unitary, angular frequency</th> <th>Fourier transform non-unitary, angular frequency</th> <th>Remarks </th></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f(x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969a95df9806ad1eb9f1c854afaba1f947d81f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:22.628ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\triangleq {\hat {f}}_{1}(\xi )\\&=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \xi x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ddf1bb65cf4f6667e4c97dd701404f26308cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.812ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{2}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f10016c4aacf29d824cab8af653e0e21157785fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.158ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\triangleq {\hat {f}}_{3}(\omega )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}}"> </td> <td>Definitions </td></tr> <tr> <td>301 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa1ae8fa7201e02499b48401ed1721b63d24d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.888ex; height:2.843ex;" alt="{\displaystyle \delta (\xi )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}\,\delta (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\,\delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ac6261db3731f0c452500736f896d59c7a2ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.121ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2\pi }}\,\delta (\omega )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \delta (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef675b4d7f05d2a0bba815c317dd2b36b3ec9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.798ex; height:2.843ex;" alt="{\displaystyle 2\pi \delta (\omega )}"> </td> <td>The distribution δ(ξ) denotes the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. </td></tr> <tr> <td>302 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d32aa9afb3c45ae68c262ffa1089d8e9143e003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.575ex; height:2.843ex;" alt="{\displaystyle \delta (x)\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2adeddfedc8362c827aca3ae9efa7a82534c7acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:5.654ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td>Dual of rule 301. </td></tr> <tr> <td>303 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{iax}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{iax}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad804a48ac2131e0928b33201545fd5a0e383ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.693ex; height:2.676ex;" alt="{\displaystyle e^{iax}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfaa5a9907b7a5f3472c495c40222927c35ce67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.412ex; height:4.843ex;" alt="{\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}\,\delta (\omega -a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\,\delta (\omega -a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88171723f408cc38a6e5bcea862247f9ec0ec4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.191ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2\pi }}\,\delta (\omega -a)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \delta (\omega -a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \delta (\omega -a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f920ee7fa7610fb8efc43751c12597c4aa9192c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle 2\pi \delta (\omega -a)}"> </td> <td>This follows from 103 and 301. </td></tr> <tr> <td>304 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3346b52ce56b4d7dacfd6628f42bf200c81b483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.48ex; height:2.843ex;" alt="{\displaystyle \cos(ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77818020961ff8051d065f665936718ee17bbc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.04ex; height:7.509ex;" alt="{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362327460f5cf628f2a71b9e63e19968c011766f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.242ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b94ca3c80dbce9a4a6017a727a4e02561858e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.117ex; height:2.843ex;" alt="{\displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)}"> </td> <td>This follows from rules 101 and 303 using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32bc4f95d1c7d0df1421d0cd3c842b040302d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.566ex; height:5.676ex;" alt="{\displaystyle \cos(ax)={\frac {e^{iax}+e^{-iax}}{2}}.}"> </td></tr> <tr> <td>305 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(ax)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(ax)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30a2961dd2164afd52abcc2f13d7d73badb41af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.224ex; height:2.843ex;" alt="{\displaystyle \sin(ax)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c5328cadbccdac20dddb5a5f9daacab0ca7ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.04ex; height:7.509ex;" alt="{\displaystyle {\frac {\delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ce01018930f51b6502c2340f225fbcbe6f071b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.242ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}\,{\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\pi {\bigl (}\delta (\omega -a)-\delta (\omega +a){\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\pi {\bigl (}\delta (\omega -a)-\delta (\omega +a){\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859efd1c6138adee06277007bf5ed8dc370c0715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.661ex; height:3.176ex;" alt="{\displaystyle -i\pi {\bigl (}\delta (\omega -a)-\delta (\omega +a){\bigr )}}"> </td> <td>This follows from 101 and 303 using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2aebcf3f2435da6664e2bd570be8cb5e65373c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.311ex; height:5.676ex;" alt="{\displaystyle \sin(ax)={\frac {e^{iax}-e^{-iax}}{2i}}.}"> </td></tr> <tr> <td>306 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(ax^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(ax^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ce1cff2a07e5f748f5b2be45142b75f36e7e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.855ex; height:3.343ex;" alt="{\displaystyle \cos \left(ax^{2}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06772eaa33b02fa7285a25b37d6850d6a6588574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.732ex; height:6.509ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>a</mi> </msqrt> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c228f28a31e9fafe2405643f6260533ec17033a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.428ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ddf1e97d0f2194579fd9e021d986e7296f8523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.756ex; height:6.509ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> </td> <td>This follows from 101 and 207 using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(ax^{2})={\frac {e^{iax^{2}}+e^{-iax^{2}}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(ax^{2})={\frac {e^{iax^{2}}+e^{-iax^{2}}}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb2373b87886338d4bbd1a43d96d3e39539a8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.284ex; height:6.009ex;" alt="{\displaystyle \cos(ax^{2})={\frac {e^{iax^{2}}+e^{-iax^{2}}}{2}}.}"> </td></tr> <tr> <td>307 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(ax^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(ax^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b8e011870f6d51b39617ccd71fda9374cf5bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.599ex; height:3.343ex;" alt="{\displaystyle \sin \left(ax^{2}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4284102942b0fec287c718cc8ca87fbc5531fec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.285ex; height:6.509ex;" alt="{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <msqrt> <mn>2</mn> <mi>a</mi> </msqrt> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bc64ccfdd567f97fc914fb1adb53c19099b559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.173ex; height:6.676ex;" alt="{\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>a</mi> </mfrac> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d601e49d8e43653f0627c8c07fe418e36cdf29f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.309ex; height:6.509ex;" alt="{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}"> </td> <td>This follows from 101 and 207 using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(ax^{2})={\frac {e^{iax^{2}}-e^{-iax^{2}}}{2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(ax^{2})={\frac {e^{iax^{2}}-e^{-iax^{2}}}{2i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185b9de6b40219504af663171d369085cb29c602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.028ex; height:6.009ex;" alt="{\displaystyle \sin(ax^{2})={\frac {e^{iax^{2}}-e^{-iax^{2}}}{2i}}.}"> </td></tr> <tr> <td>308 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi i\alpha x^{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mi>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi i\alpha x^{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0cbfc5cfcce4a056af0745fc84b6ec7d41f1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.314ex; height:3.009ex;" alt="{\displaystyle e^{-\pi i\alpha x^{2}}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\pi \xi ^{2}}{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>α</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>α</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\pi \xi ^{2}}{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639d641b1673e7dc96726453f99769e6f5f3fbb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.287ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\pi \xi ^{2}}{\alpha }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi \alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> <mi>α</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi \alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b39a6147c57f9d96b7753fee86301d5b20cf6fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.909ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi \alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>α</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301a6e25e1214b1c8e905d10938560794c188d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.415ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {\alpha }}}\,e^{-i{\frac {\pi }{4}}}e^{i{\frac {\omega ^{2}}{4\pi \alpha }}}}"> </td> <td>Here it is assumed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> is real. For the case that alpha is complex see table entry 206 above. </td></tr> <tr> <td>309 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f3d915f89551e1d02eb03af9b8b0a0a41622cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.935ex; height:2.343ex;" alt="{\displaystyle x^{n}\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87cb4b07489344585f769b71361360213cfba3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.361ex; height:6.176ex;" alt="{\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d776a55ffac33eecc15f9977ddff7affc6404fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.258ex; height:3.343ex;" alt="{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb301f0d79e1ca7a1d3b7d615d1601c01354e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.322ex; height:3.343ex;" alt="{\displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )}"> </td> <td>Here, n is a <a href="/wiki/Natural_number" title="Natural number">natural number</a> and δ(n)(ξ) is the nth distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>. </td></tr> <tr> <td>310 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(n)}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ^{(n)}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e3a1ff0aa1bc20652548e764a48e54d626cdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.69ex; height:3.343ex;" alt="{\displaystyle \delta ^{(n)}(x)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i2\pi \xi )^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i2\pi \xi )^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8589a73e3468a4b6987e30c6e06f7a9cb711d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.355ex; height:2.843ex;" alt="{\displaystyle (i2\pi \xi )^{n}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(i\omega )^{n}}{\sqrt {2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(i\omega )^{n}}{\sqrt {2\pi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ee6c574b19bcae84068e61460405ec5689049d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:6.112ex; height:6.676ex;" alt="{\displaystyle {\frac {(i\omega )^{n}}{\sqrt {2\pi }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i\omega )^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i\omega )^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f46fb8ed5e8a6ad1404c5675d4fb960ab9e011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.276ex; height:2.843ex;" alt="{\displaystyle (i\omega )^{n}}"> </td> <td>Dual of rule 309. δ(n)(ξ) is the nth distribution derivative of the Dirac delta function. This rule follows from 106 and 302. </td></tr> <tr> <td>311 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f89eaf83a3811c69adb4bf1119bafd661a4c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\pi \operatorname {sgn}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\pi \operatorname {sgn}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfab5cfc834886d4f1c688b3bc3005b56c0fef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.54ex; height:2.843ex;" alt="{\displaystyle -i\pi \operatorname {sgn}(\xi )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5e26398cebc36545cb60095f1c0db49877f06d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.116ex; height:6.343ex;" alt="{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\pi \operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\pi \operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ce5791b8ee08d2830f6b08709fbe359ae401a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.956ex; height:2.843ex;" alt="{\displaystyle -i\pi \operatorname {sgn}(\omega )}"> </td> <td>Here sgn(ξ) is the <a href="/wiki/Sign_function" title="Sign function">sign function</a>. Note that <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/x⁠ is not a distribution. It is necessary to use the <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> when testing against <a href="/wiki/Schwartz_functions" class="mw-redirect" title="Schwartz functions">Schwartz functions</a>. This rule is useful in studying the <a href="/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a>. </td></tr> <tr> <td>312 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d4f501418438356dae9c3a37d9d44ee8764def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:24.401ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\pi {\frac {(-i2\pi \xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\pi {\frac {(-i2\pi \xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bec16b4d5c255a5960cc082714ce3d0cd2339d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.64ex; height:6.676ex;" alt="{\displaystyle -i\pi {\frac {(-i2\pi \xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\,{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\,{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de00cddc208537e4840bc81bed07bf1853564731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.524ex; height:6.676ex;" alt="{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\,{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87601694edfe415336420462eadb425ac12b518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.977ex; height:6.676ex;" alt="{\displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}"> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/xn⁠ is the <a href="/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a> defined by the distributional derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fbd046db23b0faae96a6808db96f0451730adaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.904ex; height:6.676ex;" alt="{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|}"> </td></tr> <tr> <td>313 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|^{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f04c512b1f3f2fa023a6c7d6ee0f4f90fc42ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.908ex; height:3.009ex;" alt="{\displaystyle |x|^{\alpha }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>α</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43dc1bf1e41052cbd2ac55281696a8ddc081a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.762ex; height:7.343ex;" alt="{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {-2}{\sqrt {2\pi }}}\,{\frac {\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>α</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ω</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {-2}{\sqrt {2\pi }}}\,{\frac {\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec127ff3da123a91d9aa791d424b9332b14eacf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.058ex; height:7.343ex;" alt="{\displaystyle {\frac {-2}{\sqrt {2\pi }}}\,{\frac {\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>α</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ω</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcd6f243a60ed70686efac80b290f8107bd79bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.762ex; height:7.343ex;" alt="{\displaystyle -{\frac {2\sin \left({\frac {\pi \alpha }{2}}\right)\Gamma (\alpha +1)}{|\omega |^{\alpha +1}}}}"> </td> <td>This formula is valid for 0 > α > −1. For α > 0 some singular terms arise at the origin that can be found by differentiating 320. If Re α > −1, then |x|α is a locally integrable function, and so a tempered distribution. The function α ↦ |x|α is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted |x|α for α ≠ −1, −3, ... (See <a href="/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a>.) </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {|x|}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {|x|}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95fb21321232c7e90cbb324acba03b7c77558bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.783ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {|x|}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {|\xi |}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {|\xi |}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c596184c0acad38884181ece1721296cea871d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.484ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {|\xi |}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {|\omega |}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {|\omega |}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc23f885bf55932dec98c455c757a31b33e32ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.899ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {|\omega |}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {2\pi }}{\sqrt {|\omega |}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2\pi }}{\sqrt {|\omega |}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9b6aca220b242f77aeba0486ff895a32fd5cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.899ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt {2\pi }}{\sqrt {|\omega |}}}}"> </td> <td>Special case of 313. </td></tr> <tr> <td>314 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69139aa98adb4695acf691e7a04060fcd40b255c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(x)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{i\pi \xi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi>π</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{i\pi \xi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4873557be2b4f85a6924fdd36b38773cadbe347d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.001ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{i\pi \xi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9be13f4a32fd974f9b7a4c5738198cd75b50ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.576ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{i\omega }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{i\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>i</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{i\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/148ea2fe841a6e651044bd13a4e3fa99abf5aa51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.084ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{i\omega }}}"> </td> <td>The dual of rule 311. This time the Fourier transforms need to be considered as a <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a>. </td></tr> <tr> <td>315 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle u(x)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi>π</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014b632cebbe8a42b6be7ff1a1b2e3223c5ef96a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.536ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi>π</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e7dbac00766aae0a6aa1acc11b2b57b539cd72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.861ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi>π</mi> <mi>ω</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5690a870654d0a187663e5836d664cdc5cda1ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.701ex; height:6.176ex;" alt="{\displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}"> </td> <td>The function u(x) is the Heaviside <a href="/wiki/Heaviside_step_function" title="Heaviside step function">unit step function</a>; this follows from rules 101, 301, and 314. </td></tr> <tr> <td>316 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f2a50240abb18299cbaca53184b99059e193d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.633ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04545ff52d1cd0814744d791fa4fd2de72a2ebf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.503ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <mi>T</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9cf0ccf0fcfd9f40fc623e70e2024385662506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.782ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda3cea94175b1e35e4fcbd10178db9489e87f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.846ex; height:7.009ex;" alt="{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}"> </td> <td>This function is known as the <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a> function. This result can be derived from 302 and 102, together with the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sum _{n=-\infty }^{\infty }e^{inx}\\={}&2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mn>2</mn> <mi>π</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sum _{n=-\infty }^{\infty }e^{inx}\\={}&2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9555424e0a2524a5a550494dd982ea2ef4829b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:22.264ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}&\sum _{n=-\infty }^{\infty }e^{inx}\\={}&2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)\end{aligned}}}"> as distributions. </td></tr> <tr> <td>317 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{0}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{0}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cd840d13c953bbfc6368b978e6cfc263d74b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.483ex; height:2.843ex;" alt="{\displaystyle J_{0}(x)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ef776ebd01da50f4beb0f41f1f053a25edd8c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.802ex; height:7.009ex;" alt="{\displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3345a1b9adc3949fe9cd4e1b2a3abdeac8946a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.542ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\,\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\,\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13678aff482c423a56a3a0e3d1f4423eb955dd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.642ex; height:7.509ex;" alt="{\displaystyle {\frac {2\,\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> </td> <td>The function J0(x) is the zeroth order <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of first kind. </td></tr> <tr> <td>318 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3462eec70865e8698d6ae2f4a576a3bb949661e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.648ex; height:2.843ex;" alt="{\displaystyle J_{n}(x)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> <mi>rect</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812322a9acd5224b2bb53b5f97cfd052fae19b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.986ex; height:7.009ex;" alt="{\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f848e07d6632d2d41de1d345e4f5bba0e35190c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.054ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b11b628b30a5cc442f1b6339e3fe8e77acfd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.725ex; height:7.509ex;" alt="{\displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}"> </td> <td>This is a generalization of 317. The function Jn(x) is the nth order <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of first kind. The function Tn(x) is the <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomial of the first kind</a>. </td></tr> <tr> <td>319 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \left|x\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \left|x\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec0a4ec727f20be990730ae16d091d7294db928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle \log \left|x\right|}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <mi>ξ</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>γ</mi> <mi>δ</mi> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d599f3ac03cf0c6017d990a1a3fa896d9b4e5d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.344ex; height:6.009ex;" alt="{\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta \left(\xi \right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mrow> <mo>|</mo> <mi>ω</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <mi>γ</mi> <mi>δ</mi> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9b838516bbddcd564808e40a2537e1d36f60ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.97ex; height:8.509ex;" alt="{\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta \left(\omega \right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{\left|\omega \right|}}-2\pi \gamma \delta \left(\omega \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mo>|</mo> <mi>ω</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>γ</mi> <mi>δ</mi> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{\left|\omega \right|}}-2\pi \gamma \delta \left(\omega \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299d4c4c6c3b9265864219f6a352d584e7a84580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.672ex; height:5.509ex;" alt="{\displaystyle -{\frac {\pi }{\left|\omega \right|}}-2\pi \gamma \delta \left(\omega \right)}"> </td> <td>γ is the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. It is necessary to use a finite part integral when testing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/|ξ|⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/|ω|⁠against <a href="/wiki/Schwartz_functions" class="mw-redirect" title="Schwartz functions">Schwartz functions</a>. The details of this might change the coefficient of the delta function. </td></tr> <tr> <td>320 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mp ix\right)^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mo>∓</mo> <mi>i</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mp ix\right)^{-\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eaa905c2c311c09d212ac02bc9f1abeb16c7f79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:3.176ex;" alt="{\displaystyle \left(\mp ix\right)^{-\alpha }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cb439bbb2854dc959cb371062e800ad4ec445f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.208ex; height:6.509ex;" alt="{\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma \left(\alpha \right)}}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha -1}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b712e54fd61d238be311203712563874a6dadca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.588ex; height:6.676ex;" alt="{\displaystyle {\frac {\sqrt {2\pi }}{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>ω</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ee0129dfeb09d83ec680bd24d87a50d50e4e45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.588ex; height:6.009ex;" alt="{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u\left(\pm \omega \right)\left(\pm \omega \right)^{\alpha -1}}"> </td> <td>This formula is valid for 1 > α > 0. Use differentiation to derive formula for higher exponents. u is the Heaviside function. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Two-dimensional_functions">Two-dimensional functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=68" title="Edit section: Two-dimensional functions">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th></th> <th>Function</th> <th>Fourier transform unitary, ordinary frequency</th> <th>Fourier transform unitary, angular frequency</th> <th>Fourier transform non-unitary, angular frequency</th> <th>Remarks </th></tr> <tr> <td>400 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29473ed0c4e838ac9dbe074535e507166c0e9101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\triangleq \\&\iint f(x,y)e^{-i2\pi (\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\triangleq \\&\iint f(x,y)e^{-i2\pi (\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2899da4b6973722695b019903fb62469823c2ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:29.68ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\triangleq \\&\iint f(x,y)e^{-i2\pi (\xi _{x}x+\xi _{y}y)}\,dx\,dy\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&{\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&{\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94be98fbb460a4f095d0c5fff914fbb284d3c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.851ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&{\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234f53ff01216f1898f7ee08ae776bd446022ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.521ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\triangleq \\&\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy\end{aligned}}}"> </td> <td>The variables ξx, ξy, ωx, ωy are real numbers. The integrals are taken over the entire plane. </td></tr> <tr> <td>401 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4f8f3c045f9a22297609de1e9e274ec58c44cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.983ex; height:3.009ex;" alt="{\displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|ab|}}e^{-\pi \left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|ab|}}e^{-\pi \left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/905d3a1aeabf1e8d1e678b863fc183c81062227f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.348ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{|ab|}}e^{-\pi \left({\frac {\xi _{x}^{2}}{a^{2}}}+{\frac {\xi _{y}^{2}}{b^{2}}}\right)}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi \,|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi \,|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577434a7d6912a252a68154c7475b48d7b1d1ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.012ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{2\pi \,|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33d3c589a415847a3e20d60c00ab5804510ba8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.131ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{|ab|}}e^{-{\frac {1}{4\pi }}\left({\frac {\omega _{x}^{2}}{a^{2}}}+{\frac {\omega _{y}^{2}}{b^{2}}}\right)}}"> </td> <td>Both functions are Gaussians, which may not have unit volume. </td></tr> <tr> <td>402 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>circ</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7940a021169ddedca51c2f488e949cda1e6c53fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.807ex; height:6.176ex;" alt="{\displaystyle \operatorname {circ} \left({\sqrt {x^{2}+y^{2}}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065eaf8362f73d3ea644114070af6e363cefb5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:18.281ex; height:10.509ex;" alt="{\displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d25e9b6b9132a070907221a9452e6a67831214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.626ex; height:10.509ex;" alt="{\displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1191d7202f17388e29c6a4d83e774fea8bc8d46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:19.12ex; height:10.509ex;" alt="{\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> </td> <td>The function is defined by circ(r) = 1 for 0 ≤ r ≤ 1, and is 0 otherwise. The result is the amplitude distribution of the <a href="/wiki/Airy_disk" title="Airy disk">Airy disk</a>, and is expressed using J1 (the order-1 <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind).<a href="#cite_note-69">[62]</a> </td></tr> <tr> <td>403 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {x^{2}+y^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {x^{2}+y^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d46b586238687d4c8013877435bff44ebe9229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.599ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {x^{2}+y^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c395d3e2d8f6c94f3a92243e355bbc14841a19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:10.279ex; height:8.009ex;" alt="{\displaystyle {\frac {1}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0312e2f77615bd5e6676c147820b0b59705042ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:11.119ex; height:8.009ex;" alt="{\displaystyle {\frac {1}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46955e00ddb16f2217868bb636e99be3215eac66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:11.119ex; height:8.009ex;" alt="{\displaystyle {\frac {2\pi }{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}"> </td> <td>This is the <a href="/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a> of r−1, a 2-D Fourier "self-transform".<a href="#cite_note-70">[63]</a> </td></tr> <tr> <td>404 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {i}{x+iy}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {i}{x+iy}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ad47e2b3007223bbd9f83a61c6c997b39b610d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.964ex; height:5.676ex;" alt="{\displaystyle {\frac {i}{x+iy}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\xi _{x}+i\xi _{y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\xi _{x}+i\xi _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bdfbdd4e34d440a3e675e087416b64a55b24e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.738ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\xi _{x}+i\xi _{y}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\omega _{x}+i\omega _{y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\omega _{x}+i\omega _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2723a388362a0db43b70b60a8d816a0d04e7c6a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.592ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{\omega _{x}+i\omega _{y}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{\omega _{x}+i\omega _{y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{\omega _{x}+i\omega _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e25523a04f09744d3711182ad724243e955397df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.592ex; height:5.843ex;" alt="{\displaystyle {\frac {2\pi }{\omega _{x}+i\omega _{y}}}}"> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Formulas_for_general_n-dimensional_functions">Formulas for general n-dimensional functions</h3>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=69" title="Edit section: Formulas for general n-dimensional functions">edit</a>]</div> <table class="wikitable"> <tbody><tr> <th></th> <th>Function</th> <th>Fourier transform unitary, ordinary frequency</th> <th>Fourier transform unitary, angular frequency</th> <th>Fourier transform non-unitary, angular frequency</th> <th>Remarks </th></tr> <tr> <td>500 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6804844c0ec78cba3f20dfff63a8fd8693cfbaf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.886ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} )\,}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f_{1}}}({\boldsymbol {\xi }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f_{1}}}({\boldsymbol {\xi }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90f2d3c9dccaa275dbc517faf244573986f0a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.822ex; margin-bottom: -0.182ex; width:19.93ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f_{1}}}({\boldsymbol {\xi }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i2\pi {\boldsymbol {\xi }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f_{2}}}({\boldsymbol {\omega }})\triangleq \\&{\frac {1}{{(2\pi )}^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f_{2}}}({\boldsymbol {\omega }})\triangleq \\&{\frac {1}{{(2\pi )}^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b18bf17c06d5bbaa51cbdb939ba9f0a19bd2cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:25.52ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f_{2}}}({\boldsymbol {\omega }})\triangleq \\&{\frac {1}{{(2\pi )}^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f_{3}}}({\boldsymbol {\omega }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">)</mo> <mo>≜</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f_{3}}}({\boldsymbol {\omega }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd543522b92ac770c5be2a9048f015663bfb405f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.822ex; margin-bottom: -0.182ex; width:18.512ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f_{3}}}({\boldsymbol {\omega }})\triangleq \\&\int _{\mathbb {R} ^{n}}f(\mathbf {x} )e^{-i{\boldsymbol {\omega }}\cdot \mathbf {x} }\,d\mathbf {x} \end{aligned}}}"> </td> <td> </td></tr> <tr> <td>501 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{[0,1]}(|\mathbf {x} |)\left(1-|\mathbf {x} |^{2}\right)^{\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{[0,1]}(|\mathbf {x} |)\left(1-|\mathbf {x} |^{2}\right)^{\delta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f3e6c1f52f0ee9366c40ee6f67c65f11dbd03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.728ex; height:5.343ex;" alt="{\displaystyle \chi _{[0,1]}(|\mathbf {x} |)\left(1-|\mathbf {x} |^{2}\right)^{\delta }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }\,|{\boldsymbol {\xi }}|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(2\pi |{\boldsymbol {\xi }}|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }\,|{\boldsymbol {\xi }}|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(2\pi |{\boldsymbol {\xi }}|)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a595c322b60e1d89d10bd55f0cfa60bcf136f022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.853ex; height:7.343ex;" alt="{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }\,|{\boldsymbol {\xi }}|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(2\pi |{\boldsymbol {\xi }}|)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\delta }\,{\frac {\Gamma (\delta +1)}{\left|{\boldsymbol {\omega }}\right|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(|{\boldsymbol {\omega }}|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> </mrow> </msup> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\delta }\,{\frac {\Gamma (\delta +1)}{\left|{\boldsymbol {\omega }}\right|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(|{\boldsymbol {\omega }}|)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0807ac06201171751e53ca547eaee058f3767e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.624ex; height:7.343ex;" alt="{\displaystyle 2^{\delta }\,{\frac {\Gamma (\delta +1)}{\left|{\boldsymbol {\omega }}\right|^{{\frac {n}{2}}+\delta }}}J_{{\frac {n}{2}}+\delta }(|{\boldsymbol {\omega }}|)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }}}\left|{\frac {\boldsymbol {\omega }}{2\pi }}\right|^{-{\frac {n}{2}}-\delta }J_{{\frac {n}{2}}+\delta }(\!|{\boldsymbol {\omega }}|\!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold-italic">ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>δ</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="negativethinmathspace" /> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }}}\left|{\frac {\boldsymbol {\omega }}{2\pi }}\right|^{-{\frac {n}{2}}-\delta }J_{{\frac {n}{2}}+\delta }(\!|{\boldsymbol {\omega }}|\!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76dd95223605db27c11603e15eb0da9ca588455e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.119ex; height:6.009ex;" alt="{\displaystyle {\frac {\Gamma (\delta +1)}{\pi ^{\delta }}}\left|{\frac {\boldsymbol {\omega }}{2\pi }}\right|^{-{\frac {n}{2}}-\delta }J_{{\frac {n}{2}}+\delta }(\!|{\boldsymbol {\omega }}|\!)}"> </td> <td>The function χ[0, 1] is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of the interval [0, 1]. The function Γ(x) is the gamma function. The function J<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/2⁠ + δ is a Bessel function of the first kind, with order <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/2⁠ + δ. Taking n = 2 and δ = 0 produces 402.<a href="#cite_note-71">[64]</a> </td></tr> <tr> <td>502 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {x} |^{-\alpha },\quad 0<\operatorname {Re} \alpha <n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>Re</mi> <mo>⁡</mo> <mi>α</mi> <mo><</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {x} |^{-\alpha },\quad 0<\operatorname {Re} \alpha <n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55904386b79dd5115c8e9ab7166c5346ae783f23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.643ex; height:3.176ex;" alt="{\displaystyle |\mathbf {x} |^{-\alpha },\quad 0<\operatorname {Re} \alpha <n.}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(2\pi )^{\alpha }}{c_{n,\alpha }}}|{\boldsymbol {\xi }}|^{-(n-\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(2\pi )^{\alpha }}{c_{n,\alpha }}}|{\boldsymbol {\xi }}|^{-(n-\alpha )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f934c42cbe93a1f80d3ca9e28295d74599ae04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.006ex; height:6.343ex;" alt="{\displaystyle {\frac {(2\pi )^{\alpha }}{c_{n,\alpha }}}|{\boldsymbol {\xi }}|^{-(n-\alpha )}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(2\pi )^{\frac {n}{2}}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(2\pi )^{\frac {n}{2}}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4817d3c29a8e2dc2ad5217dcec637ff1ce90984e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.078ex; height:7.176ex;" alt="{\displaystyle {\frac {(2\pi )^{\frac {n}{2}}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(2\pi )^{n}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(2\pi )^{n}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8f1daf526cba1892b45ac3060bf79b27991302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.428ex; height:6.343ex;" alt="{\displaystyle {\frac {(2\pi )^{n}}{c_{n,\alpha }}}|{\boldsymbol {\omega }}|^{-(n-\alpha )}}"> </td> <td>See <a href="/wiki/Riesz_potential" title="Riesz potential">Riesz potential</a> where the constant is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n,\alpha }=\pi ^{\frac {n}{2}}2^{\alpha }{\frac {\Gamma \left({\frac {\alpha }{2}}\right)}{\Gamma \left({\frac {n-\alpha }{2}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>α</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n,\alpha }=\pi ^{\frac {n}{2}}2^{\alpha }{\frac {\Gamma \left({\frac {\alpha }{2}}\right)}{\Gamma \left({\frac {n-\alpha }{2}}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe860f695602037bbe5189bba5fbaadcb748728f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.088ex; height:7.676ex;" alt="{\displaystyle c_{n,\alpha }=\pi ^{\frac {n}{2}}2^{\alpha }{\frac {\Gamma \left({\frac {\alpha }{2}}\right)}{\Gamma \left({\frac {n-\alpha }{2}}\right)}}.}"> The formula also holds for all α ≠ n, n + 2, ... by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See <a href="/wiki/Homogeneous_distribution" title="Homogeneous distribution">homogeneous distribution</a>.<a href="#cite_note-72">[note 8]</a> </td></tr> <tr> <td>503 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\left|{\boldsymbol {\sigma }}\right|\left(2\pi \right)^{\frac {n}{2}}}}e^{-{\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }{\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}\mathbf {x} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>|</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\left|{\boldsymbol {\sigma }}\right|\left(2\pi \right)^{\frac {n}{2}}}}e^{-{\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }{\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}\mathbf {x} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a963d44dec0f15b8a0401661c290b63d5a7284f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.795ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\left|{\boldsymbol {\sigma }}\right|\left(2\pi \right)^{\frac {n}{2}}}}e^{-{\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }{\boldsymbol {\sigma }}^{-\mathrm {T} }{\boldsymbol {\sigma }}^{-1}\mathbf {x} }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-2\pi ^{2}{\boldsymbol {\xi }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\xi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-2\pi ^{2}{\boldsymbol {\xi }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\xi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8d4433cb9705bd48f188c5124edaf24feb3362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.372ex; height:3.176ex;" alt="{\displaystyle e^{-2\pi ^{2}{\boldsymbol {\xi }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\xi }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2\pi )^{-{\frac {n}{2}}}e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2\pi )^{-{\frac {n}{2}}}e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e793fc7fe30b21bd87aff3ca525fbb81aea6fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.419ex; height:4.009ex;" alt="{\displaystyle (2\pi )^{-{\frac {n}{2}}}e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3249ae8c9606a681192bbb36cdf0bcbe3e9068b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.968ex; height:3.509ex;" alt="{\displaystyle e^{-{\frac {1}{2}}{\boldsymbol {\omega }}^{\mathrm {T} }{\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{\mathrm {T} }{\boldsymbol {\omega }}}}"> </td> <td>This is the formula for a <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page, Σ = σ σT and Σ−1 = σ−T σ−1 </td></tr> <tr> <td>504 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-2\pi \alpha |\mathbf {x} |}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-2\pi \alpha |\mathbf {x} |}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d7ee7b274c84c901fbb259abcdb5a51659d61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.843ex;" alt="{\displaystyle e^{-2\pi \alpha |\mathbf {x} |}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c_{n}\alpha }{\left(\alpha ^{2}+|{\boldsymbol {\xi }}|^{2}\right)^{\frac {n+1}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>α</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c_{n}\alpha }{\left(\alpha ^{2}+|{\boldsymbol {\xi }}|^{2}\right)^{\frac {n+1}{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2120868a79ff1d62a7e0bcc8ff3165e5f9b8931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:16.097ex; height:8.509ex;" alt="{\displaystyle {\frac {c_{n}\alpha }{\left(\alpha ^{2}+|{\boldsymbol {\xi }}|^{2}\right)^{\frac {n+1}{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c_{n}(2\pi )^{\frac {n+2}{2}}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi>α</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c_{n}(2\pi )^{\frac {n+2}{2}}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5075c88d64b73eb060137055874cc04281525b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:20.136ex; height:10.843ex;" alt="{\displaystyle {\frac {c_{n}(2\pi )^{\frac {n+2}{2}}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c_{n}(2\pi )^{n+1}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>α</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c_{n}(2\pi )^{n+1}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b18ba76f3aa9a3b7021a73b34350927249b81b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:20.136ex; height:9.676ex;" alt="{\displaystyle {\frac {c_{n}(2\pi )^{n+1}\alpha }{\left(4\pi ^{2}\alpha ^{2}+|{\boldsymbol {\omega }}|^{2}\right)^{\frac {n+1}{2}}}}}"> </td> <td>Here<a href="#cite_note-73">[65]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\pi ^{\frac {n+1}{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\pi ^{\frac {n+1}{2}}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19086748c5f00add34d5a41e1c9bef9915cef136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.345ex; height:9.009ex;" alt="{\displaystyle c_{n}={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\pi ^{\frac {n+1}{2}}}},}"> Re(α) > 0 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=70" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Analog_signal_processing" title="Analog signal processing">Analog signal processing</a></li> <li><a href="/wiki/Beevers%E2%80%93Lipson_strip" title="Beevers–Lipson strip">Beevers–Lipson strip</a></li> <li><a href="/wiki/Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li> <li><a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">DFT matrix</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transform</a></li> <li><a href="/wiki/Fourier_integral_operator" title="Fourier integral operator">Fourier integral operator</a></li> <li><a href="/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">Fourier inversion theorem</a></li> <li><a href="/wiki/Fourier_multiplier" class="mw-redirect" title="Fourier multiplier">Fourier multiplier</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Fourier_sine_transform" class="mw-redirect" title="Fourier sine transform">Fourier sine transform</a></li> <li><a href="/wiki/Fourier%E2%80%93Deligne_transform" title="Fourier–Deligne transform">Fourier–Deligne transform</a></li> <li><a href="/wiki/Fourier%E2%80%93Mukai_transform" title="Fourier–Mukai transform">Fourier–Mukai transform</a></li> <li><a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></li> <li><a href="/wiki/Indirect_Fourier_transform" class="mw-redirect" title="Indirect Fourier transform">Indirect Fourier transform</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a> <ul><li><a href="/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a></li> <li><a href="/wiki/Hartley_transform" title="Hartley transform">Hartley transform</a></li></ul></li> <li><a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Linear_canonical_transform" class="mw-redirect" title="Linear canonical transform">Linear canonical transform</a></li> <li><a href="/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">List of Fourier-related transforms</a></li> <li><a href="/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a></li> <li><a href="/wiki/Multidimensional_transform" title="Multidimensional transform">Multidimensional transform</a></li> <li><a href="/wiki/NGC_4622" title="NGC 4622">NGC 4622</a>, especially the image NGC 4622 Fourier transform m = 2.</li> <li><a href="/wiki/Nonlocal_operator" title="Nonlocal operator">Nonlocal operator</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quadratic_Fourier_transform" title="Quadratic Fourier transform">Quadratic Fourier transform</a></li> <li><a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">Short-time Fourier transform</a></li> <li><a href="/wiki/Spectral_density" title="Spectral density">Spectral density</a> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li></ul></li> <li><a href="/wiki/Symbolic_integration" title="Symbolic integration">Symbolic integration</a></li> <li><a href="/wiki/Time_stretch_dispersive_Fourier_transform" title="Time stretch dispersive Fourier transform">Time stretch dispersive Fourier transform</a></li> <li><a href="/wiki/Transform_(mathematics)" class="mw-redirect" title="Transform (mathematics)">Transform (mathematics)</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=71" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Depending on the application a <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>, <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributional</a>, or other approach may be most appropriate. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFVretblad2000">Vretblad (2000)</a> provides solid justification for these formal procedures without going too deeply into <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> or the <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">theory of distributions</a>. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> In <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> one encounters vector-valued Fourier transforms of multi-component wave functions. In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for example <a href="#CITEREFGreinerReinhardt1996">Greiner & Reinhardt (1996)</a>. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> A possible source of confusion is the <a href="#Frequency_shifting">frequency-shifting property</a>; i.e. the transform of function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)e^{-i2\pi \xi _{0}x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)e^{-i2\pi \xi _{0}x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783c0a7783daa653cc12b6feea7e173c51ca30a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.835ex; height:3.176ex;" alt="{\displaystyle f(x)e^{-i2\pi \xi _{0}x}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi +\xi _{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>+</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi +\xi _{0}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d564de4ba3f83e1e0a141b69175e17423313901c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.17ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi +\xi _{0}).}">  The value of this function at  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5354e193004a0e2f16e7d4a76ea499ffcca225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.291ex; height:2.509ex;" alt="{\displaystyle \xi =0}">  is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(\xi _{0}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(\xi _{0}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db49aeb991a6d6036cd9b50881e7e4b1f94c085e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.3ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(\xi _{0}),}"> meaning that a frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d62e210399a8a9c64f9c534597f2acd23f2a1f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.073ex; height:2.509ex;" alt="{\displaystyle \xi _{0}}"> has been shifted to zero (also see <a href="/wiki/Negative_frequency#Simplifying_the_Fourier_transform" title="Negative frequency">Negative frequency</a>). </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> The operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b3b1bd8becc97a2bd471cf90e4f3ca0aaf888a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.303ex; height:6.176ex;" alt="{\displaystyle U\left({\frac {1}{2\pi }}{\frac {d}{dx}}\right)}"> is defined by replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}{\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}{\frac {d}{dx}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b665c6ee1e065362b8ce50f9754d27af61b7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.712ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{2\pi }}{\frac {d}{dx}}}"> in the <a href="/wiki/Taylor_series" title="Taylor series">Taylor expansion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d8601ed3faf8353e5d0b6c97540b46c9a4ce52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.568ex; height:2.843ex;" alt="{\displaystyle U(x).}"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> Up to an imaginary constant factor whose magnitude depends on what Fourier transform convention is used. </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> The direct command <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fourier transform of cos(6*pi*t) exp(−pi*t^2)</code> would also work for Wolfram Alpha, although the options for the convention (see <a href="#Other_conventions">Fourier transform § Other conventions</a>) must be changed away from the default option, which is actually equivalent to <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">integrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf</code>. </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> In <a href="#CITEREFGelfandShilov1964">Gelfand & Shilov 1964</a>, p. 363, with the non-unitary conventions of this table, the transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {x} |^{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {x} |^{\lambda }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8787490b5a8753a87ae7ae60999cc6d3537357c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.895ex; height:3.343ex;" alt="{\displaystyle |\mathbf {x} |^{\lambda }}"> is given to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\lambda +n}\pi ^{{\tfrac {1}{2}}n}{\frac {\Gamma \left({\frac {\lambda +n}{2}}\right)}{\Gamma \left(-{\frac {\lambda }{2}}\right)}}|{\boldsymbol {\omega }}|^{-\lambda -n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mo>+</mo> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\lambda +n}\pi ^{{\tfrac {1}{2}}n}{\frac {\Gamma \left({\frac {\lambda +n}{2}}\right)}{\Gamma \left(-{\frac {\lambda }{2}}\right)}}|{\boldsymbol {\omega }}|^{-\lambda -n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c751fd6d92b9ade3f5a627cb6270189add38edfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:26.036ex; height:10.176ex;" alt="{\displaystyle 2^{\lambda +n}\pi ^{{\tfrac {1}{2}}n}{\frac {\Gamma \left({\frac {\lambda +n}{2}}\right)}{\Gamma \left(-{\frac {\lambda }{2}}\right)}}|{\boldsymbol {\omega }}|^{-\lambda -n}}"> from which this follows, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =-\alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f086450e87d9c85c6e9611e73cdd6149f496179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.749ex; height:2.343ex;" alt="{\displaystyle \lambda =-\alpha }">. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=72" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 22em;"> <ol class="references"> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFKhareButolaRajora2023">Khare, Butola & Rajora 2023</a>, pp. 13–14 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFKaiser1994">Kaiser 1994</a>, p. 29 </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFRahman2011">Rahman 2011</a>, p. 11 </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFDymMcKean1985">Dym & McKean 1985</a> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFFourier1822">Fourier 1822</a>, p. 525 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFFourier1878">Fourier 1878</a>, p. 408 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFJordan1883">Jordan (1883)</a> proves on pp. 216–226 the <a href="/wiki/Fourier_inversion_theorem#Fourier_integral_theorem" title="Fourier inversion theorem">Fourier integral theorem</a> before studying Fourier series. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFTitchmarsh1986">Titchmarsh 1986</a>, p. 1 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFRahman2011">Rahman 2011</a>, p. 10. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFOppenheimSchaferBuck1999">Oppenheim, Schafer & Buck 1999</a>, p. 58 </li> <li id="cite_note-Stein-Weiss-1971-15">^ <a href="#cite_ref-Stein-Weiss-1971_15-0">a</a> <a href="#cite_ref-Stein-Weiss-1971_15-1">b</a> <a href="#cite_ref-Stein-Weiss-1971_15-2">c</a> <a href="#cite_ref-Stein-Weiss-1971_15-3">d</a> <a href="#cite_ref-Stein-Weiss-1971_15-4">e</a> <a href="#cite_ref-Stein-Weiss-1971_15-5">f</a> <a href="#cite_ref-Stein-Weiss-1971_15-6">g</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFFolland1989">Folland 1989</a> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFFourier1822">Fourier 1822</a> </li> <li id="cite_note-Pinsky-2002-18">^ <a href="#cite_ref-Pinsky-2002_18-0">a</a> <a href="#cite_ref-Pinsky-2002_18-1">b</a> <a href="#cite_ref-Pinsky-2002_18-2">c</a> <a href="#cite_ref-Pinsky-2002_18-3">d</a> <a href="#cite_ref-Pinsky-2002_18-4">e</a> <a href="#CITEREFPinsky2002">Pinsky 2002</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFArfken1985">Arfken 1985</a> </li> <li id="cite_note-Katznelson-1976-20">^ <a href="#cite_ref-Katznelson-1976_20-0">a</a> <a href="#cite_ref-Katznelson-1976_20-1">b</a> <a href="#cite_ref-Katznelson-1976_20-2">c</a> <a href="#cite_ref-Katznelson-1976_20-3">d</a> <a href="#cite_ref-Katznelson-1976_20-4">e</a> <a href="#CITEREFKatznelson1976">Katznelson 1976</a> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, p. 187 </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, p. 186 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFFolland1992">Folland 1992</a>, p. 216 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFWolf1979">Wolf 1979</a>, p. 307ff </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFFolland1989">Folland 1989</a>, p. 53 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFCeleghiniGadelladel_Olmo2021">Celeghini, Gadella & del Olmo 2021</a> </li> <li id="cite_note-Duoandikoetxea-2001-28">^ <a href="#cite_ref-Duoandikoetxea-2001_28-0">a</a> <a href="#cite_ref-Duoandikoetxea-2001_28-1">b</a> <a href="#CITEREFDuoandikoetxea2001">Duoandikoetxea 2001</a> </li> <li id="cite_note-Boashash-2003-29">^ <a href="#cite_ref-Boashash-2003_29-0">a</a> <a href="#cite_ref-Boashash-2003_29-1">b</a> <a href="#CITEREFBoashash2003">Boashash 2003</a> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFCondon1937">Condon 1937</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <a href="#CITEREFWolf1979">Wolf 1979</a>, p. 320 </li> <li id="cite_note-auto-32">^ <a href="#cite_ref-auto_32-0">a</a> <a href="#cite_ref-auto_32-1">b</a> <a href="#CITEREFWolf1979">Wolf 1979</a>, p. 312 </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <a href="#CITEREFFolland1989">Folland 1989</a>, p. 52 </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a href="#CITEREFHowe1980">Howe 1980</a> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFPaleyWiener1934">Paley & Wiener 1934</a> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <a href="#CITEREFGelfandVilenkin1964">Gelfand & Vilenkin 1964</a> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFKirillovGvishiani1982">Kirillov & Gvishiani 1982</a> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFClozelDelorme1985">Clozel & Delorme 1985</a>, pp. 331–333 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFde_GrootMazur1984">de Groot & Mazur 1984</a>, p. 146 </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFChampeney1987">Champeney 1987</a>, p. 80 </li> <li id="cite_note-Kolmogorov-Fomin-1999-41">^ <a href="#cite_ref-Kolmogorov-Fomin-1999_41-0">a</a> <a href="#cite_ref-Kolmogorov-Fomin-1999_41-1">b</a> <a href="#cite_ref-Kolmogorov-Fomin-1999_41-2">c</a> <a href="#CITEREFKolmogorovFomin1999">Kolmogorov & Fomin 1999</a> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="#CITEREFWiener1949">Wiener 1949</a> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <a href="#CITEREFChampeney1987">Champeney 1987</a>, p. 63 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <a href="#CITEREFWidderWiener1938">Widder & Wiener 1938</a>, p. 537 </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a href="#CITEREFPinsky2002">Pinsky 2002</a>, p. 131 </li> <li id="cite_note-Stein-Shakarchi-2003-46"><a href="#cite_ref-Stein-Shakarchi-2003_46-0">^</a> <a href="#CITEREFSteinShakarchi2003">Stein & Shakarchi 2003</a> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <a href="#CITEREFSteinShakarchi2003">Stein & Shakarchi 2003</a>, p. 158 </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <a href="#CITEREFChatfield2004">Chatfield 2004</a>, p. 113 </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <a href="#CITEREFFourier1822">Fourier 1822</a>, p. 441 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a href="#CITEREFPoincaré1895">Poincaré 1895</a>, p. 102 </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a href="#CITEREFWhittakerWatson1927">Whittaker & Watson 1927</a>, p. 188 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <a href="#CITEREFGrafakos2004">Grafakos 2004</a> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFGrafakosTeschl2013">Grafakos & Teschl 2013</a> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> More generally, one can take a sequence of functions that are in the intersection of L1 and L2 and that converges to f in the L2-norm, and define the Fourier transform of f as the L2 -limit of the Fourier transforms of these functions. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://statweb.stanford.edu/~candes/teaching/math262/Lectures/Lecture03.pdf">"Applied Fourier Analysis and Elements of Modern Signal Processing Lecture 3"</a> (PDF). 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Retrieved 2019-10-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Applied+Fourier+Analysis+and+Elements+of+Modern+Signal+Processing+Lecture+3&rft.date=2016-01-12&rft_id=https%3A%2F%2Fstatweb.stanford.edu%2F~candes%2Fteaching%2Fmath262%2FLectures%2FLecture03.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, Thm. 2.3 </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFPinsky2002">Pinsky 2002</a>, p. 256 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <a href="#CITEREFHewittRoss1970">Hewitt & Ross 1970</a>, Chapter 8 </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <a href="#CITEREFKnapp2001">Knapp 2001</a> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorreiaJustoAngélico2024" class="citation journal cs1">Correia, L. 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The Fourier Transform .com. 2015 [2010]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220126171340/https://www.thefouriertransform.com/transform/integration.php">Archived</a> from the original on 2022-01-26. Retrieved 2023-08-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Fourier+Transform+.com&rft.atitle=The+Integration+Property+of+the+Fourier+Transform&rft.date=2015&rft_id=https%3A%2F%2Fwww.thefouriertransform.com%2Ftransform%2Fintegration.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, Thm. 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(1987), A Handbook of Fourier Theorems, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Handbook+of+Fourier+Theorems&rft.pub=Cambridge+University+Press&rft.date=1987&rft.aulast=Champeney&rft.aufirst=D.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChatfield2004" class="citation cs2">Chatfield, Chris (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qKzyAbdaDFAC&q=%22Fourier+transform%22">The Analysis of Time Series: An Introduction</a>, Texts in Statistical Science (6th ed.), London: Chapman & Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780203491683" title="Special:BookSources/9780203491683"><bdi>9780203491683</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analysis+of+Time+Series%3A+An+Introduction&rft.place=London&rft.series=Texts+in+Statistical+Science&rft.edition=6th&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2004&rft.isbn=9780203491683&rft.aulast=Chatfield&rft.aufirst=Chris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqKzyAbdaDFAC%26q%3D%2522Fourier%2Btransform%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClozelDelorme1985" class="citation cs2">Clozel, Laurent; Delorme, Patrice (1985), "Sur le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs réels", Comptes Rendus de l'Académie des Sciences, Série I, 300: 331–333</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+Rendus+de+l%27Acad%C3%A9mie+des+Sciences%2C+S%C3%A9rie+I&rft.atitle=Sur+le+th%C3%A9or%C3%A8me+de+Paley-Wiener+invariant+pour+les+groupes+de+Lie+r%C3%A9ductifs+r%C3%A9els&rft.volume=300&rft.pages=331-333&rft.date=1985&rft.aulast=Clozel&rft.aufirst=Laurent&rft.au=Delorme%2C+Patrice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCondon1937" class="citation cs2"><a href="/wiki/Edward_Condon" title="Edward Condon">Condon, E. U.</a> (1937), "Immersion of the Fourier transform in a continuous group of functional transformations", <a href="/wiki/PNAS" class="mw-redirect" title="PNAS">Proc. Natl. Acad. 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(1985), Fourier Series and Integrals, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-226451-1" title="Special:BookSources/978-0-12-226451-1"><bdi>978-0-12-226451-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Series+and+Integrals&rft.pub=Academic+Press&rft.date=1985&rft.isbn=978-0-12-226451-1&rft.aulast=Dym&rft.aufirst=H.&rft.au=McKean%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEaston2010" class="citation cs2">Easton, Roger L. 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Joseph</a> (1822), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ&q=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&pg=PA525">Théorie analytique de la chaleur</a> (in French), Paris: Firmin Didot, père et fils, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2688081">2688081</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.place=Paris&rft.pub=Firmin+Didot%2C+p%C3%A8re+et+fils&rft.date=1822&rft_id=info%3Aoclcnum%2F2688081&rft.aulast=Fourier&rft.aufirst=J.B.+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTDQJAAAAIAAJ%26q%3D%2522c%2527est-%25C3%25A0-dire%2Bqu%2527on%2Ba%2Bl%2527%25C3%25A9quation%2522%26pg%3DPA525&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1878" class="citation cs2"><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier, J.B. Joseph</a> (1878) [1822], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-N8EAAAAYAAJ&q=%22that+is+to+say%2C+that+we+have+the+equation%22&pg=PA408">The Analytical Theory of Heat</a>, translated by Alexander Freeman, The University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analytical+Theory+of+Heat&rft.pub=The+University+Press&rft.date=1878&rft.aulast=Fourier&rft.aufirst=J.B.+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-N8EAAAAYAAJ%26q%3D%2522that%2Bis%2Bto%2Bsay%252C%2Bthat%2Bwe%2Bhave%2Bthe%2Bequation%2522%26pg%3DPA408&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> (translated from French)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGradshteynRyzhikGeronimusTseytlin2015" class="citation cs2"><a href="/wiki/Izrail_Solomonovich_Gradshteyn" class="mw-redirect" title="Izrail Solomonovich Gradshteyn">Gradshteyn, Izrail Solomonovich</a>; <a href="/wiki/Iosif_Moiseevich_Ryzhik" class="mw-redirect" title="Iosif Moiseevich Ryzhik">Ryzhik, Iosif Moiseevich</a>; <a href="/wiki/Yuri_Veniaminovich_Geronimus" class="mw-redirect" title="Yuri Veniaminovich Geronimus">Geronimus, Yuri Veniaminovich</a>; <a href="/wiki/Michail_Yulyevich_Tseytlin" class="mw-redirect" title="Michail Yulyevich Tseytlin">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015), Zwillinger, Daniel; <a href="/wiki/Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (eds.), <a href="/wiki/Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Table of Integrals, Series, and Products</a>, translated by Scripta Technica, Inc. (8th ed.), <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-384933-5" title="Special:BookSources/978-0-12-384933-5"><bdi>978-0-12-384933-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Table+of+Integrals%2C+Series%2C+and+Products&rft.edition=8th&rft.pub=Academic+Press&rft.date=2015&rft.isbn=978-0-12-384933-5&rft.aulast=Gradshteyn&rft.aufirst=Izrail+Solomonovich&rft.au=Ryzhik%2C+Iosif+Moiseevich&rft.au=Geronimus%2C+Yuri+Veniaminovich&rft.au=Tseytlin%2C+Michail+Yulyevich&rft.au=Jeffrey%2C+Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrafakos2004" class="citation cs2">Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Prentice-Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-035399-3" title="Special:BookSources/978-0-13-035399-3"><bdi>978-0-13-035399-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+and+Modern+Fourier+Analysis&rft.pub=Prentice-Hall&rft.date=2004&rft.isbn=978-0-13-035399-3&rft.aulast=Grafakos&rft.aufirst=Loukas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrafakosTeschl2013" class="citation cs2">Grafakos, Loukas; <a href="/wiki/Gerald_Teschl" title="Gerald Teschl">Teschl, Gerald</a> (2013), "On Fourier transforms of radial functions and distributions", J. Fourier Anal. Appl., 19: 167–179, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.5469">1112.5469</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00041-012-9242-5">10.1007/s00041-012-9242-5</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1280745">1280745</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Fourier+Anal.+Appl.&rft.atitle=On+Fourier+transforms+of+radial+functions+and+distributions&rft.volume=19&rft.pages=167-179&rft.date=2013&rft_id=info%3Aarxiv%2F1112.5469&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1280745%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00041-012-9242-5&rft.aulast=Grafakos&rft.aufirst=Loukas&rft.au=Teschl%2C+Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreinerReinhardt1996" class="citation cs2">Greiner, W.; Reinhardt, J. (1996), <a rel="nofollow" class="external text" href="https://archive.org/details/fieldquantizatio0000grei">Field Quantization</a>, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-59179-5" title="Special:BookSources/978-3-540-59179-5"><bdi>978-3-540-59179-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Field+Quantization&rft.pub=Springer&rft.date=1996&rft.isbn=978-3-540-59179-5&rft.aulast=Greiner&rft.aufirst=W.&rft.au=Reinhardt%2C+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffieldquantizatio0000grei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandShilov1964" class="citation cs2"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="/wiki/Naum_Ya._Vilenkin" class="mw-redirect" title="Naum Ya. Vilenkin">Shilov, G.E.</a> (1964), Generalized Functions, vol. 1, New York: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Functions&rft.place=New+York&rft.pub=Academic+Press&rft.date=1964&rft.aulast=Gelfand&rft.aufirst=I.M.&rft.au=Shilov%2C+G.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> (translated from Russian)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelfandVilenkin1964" class="citation cs2"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="/wiki/Naum_Ya._Vilenkin" class="mw-redirect" title="Naum Ya. Vilenkin">Vilenkin, N.Y.</a> (1964), Generalized Functions, vol. 4, New York: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Functions&rft.place=New+York&rft.pub=Academic+Press&rft.date=1964&rft.aulast=Gelfand&rft.aufirst=I.M.&rft.au=Vilenkin%2C+N.Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> (translated from Russian)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHewittRoss1970" class="citation cs2 cs1-prop-long-vol">Hewitt, Edwin; Ross, Kenneth A. (1970), Abstract harmonic analysis, Die Grundlehren der mathematischen Wissenschaften, Band 152, vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0262773">0262773</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+harmonic+analysis&rft.series=Die+Grundlehren+der+mathematischen+Wissenschaften%2C+Band+152&rft.pub=Springer&rft.date=1970&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0262773%23id-name%3DMR&rft.aulast=Hewitt&rft.aufirst=Edwin&rft.au=Ross%2C+Kenneth+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHörmander1976" class="citation cs2"><a href="/wiki/Lars_H%C3%B6rmander" title="Lars Hörmander">Hörmander, L.</a> (1976), Linear Partial Differential Operators, vol. 1, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-00662-6" title="Special:BookSources/978-3-540-00662-6"><bdi>978-3-540-00662-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Partial+Differential+Operators&rft.pub=Springer&rft.date=1976&rft.isbn=978-3-540-00662-6&rft.aulast=H%C3%B6rmander&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowe1980" class="citation cs2">Howe, Roger (1980), "On the role of the Heisenberg group in harmonic analysis", <a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>, 3 (2): 821–844, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14825-9">10.1090/S0273-0979-1980-14825-9</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0578375">0578375</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=On+the+role+of+the+Heisenberg+group+in+harmonic+analysis&rft.volume=3&rft.issue=2&rft.pages=821-844&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1980-14825-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D578375%23id-name%3DMR&rft.aulast=Howe&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames2011" class="citation cs2">James, J.F. (2011), A Student's Guide to Fourier Transforms (3rd ed.), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-17683-5" title="Special:BookSources/978-0-521-17683-5"><bdi>978-0-521-17683-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Student%27s+Guide+to+Fourier+Transforms&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-0-521-17683-5&rft.aulast=James&rft.aufirst=J.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJordan1883" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1883), Cours d'Analyse de l'École Polytechnique, vol. II, Calcul Intégral: Intégrales définies et indéfinies (2nd ed.), Paris</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+d%27Analyse+de+l%27%C3%89cole+Polytechnique&rft.place=Paris&rft.edition=2nd&rft.date=1883&rft.aulast=Jordan&rft.aufirst=Camille&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaiser1994" class="citation cs2">Kaiser, Gerald (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rfRnrhJwoloC&q=%22becomes+the+Fourier+%28integral%29+transform%22&pg=PA29">"A Friendly Guide to Wavelets"</a>, Physics Today, 48 (7): 57–58, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PhT....48g..57K">1995PhT....48g..57K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2808105">10.1063/1.2808105</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3711-8" title="Special:BookSources/978-0-8176-3711-8"><bdi>978-0-8176-3711-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=A+Friendly+Guide+to+Wavelets&rft.volume=48&rft.issue=7&rft.pages=57-58&rft.date=1994&rft_id=info%3Adoi%2F10.1063%2F1.2808105&rft_id=info%3Abibcode%2F1995PhT....48g..57K&rft.isbn=978-0-8176-3711-8&rft.aulast=Kaiser&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrfRnrhJwoloC%26q%3D%2522becomes%2Bthe%2BFourier%2B%2528integral%2529%2Btransform%2522%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKammler2000" class="citation cs2">Kammler, David (2000), A First Course in Fourier Analysis, Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-578782-3" title="Special:BookSources/978-0-13-578782-3"><bdi>978-0-13-578782-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Fourier+Analysis&rft.pub=Prentice+Hall&rft.date=2000&rft.isbn=978-0-13-578782-3&rft.aulast=Kammler&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatznelson1976" class="citation cs2">Katznelson, Yitzhak (1976), An Introduction to Harmonic Analysis, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63331-2" title="Special:BookSources/978-0-486-63331-2"><bdi>978-0-486-63331-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Harmonic+Analysis&rft.pub=Dover&rft.date=1976&rft.isbn=978-0-486-63331-2&rft.aulast=Katznelson&rft.aufirst=Yitzhak&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhareButolaRajora2023" class="citation cs2">Khare, Kedar; Butola, Mansi; Rajora, Sunaina (2023), "Chapter 2.3 Fourier Transform as a Limiting Case of Fourier Series", Fourier Optics and Computational Imaging (2nd ed.), Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-18353-9">10.1007/978-3-031-18353-9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-18353-9" title="Special:BookSources/978-3-031-18353-9"><bdi>978-3-031-18353-9</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:255676773">255676773</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+2.3+Fourier+Transform+as+a+Limiting+Case+of+Fourier+Series&rft.btitle=Fourier+Optics+and+Computational+Imaging&rft.edition=2nd&rft.pub=Springer&rft.date=2023&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A255676773%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-031-18353-9&rft.isbn=978-3-031-18353-9&rft.aulast=Khare&rft.aufirst=Kedar&rft.au=Butola%2C+Mansi&rft.au=Rajora%2C+Sunaina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirillovGvishiani1982" class="citation cs2"><a href="/wiki/Alexandre_Kirillov" title="Alexandre Kirillov">Kirillov, Alexandre</a>; Gvishiani, Alexei D. (1982) [1979], Theorems and Problems in Functional Analysis, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorems+and+Problems+in+Functional+Analysis&rft.pub=Springer&rft.date=1982&rft.aulast=Kirillov&rft.aufirst=Alexandre&rft.au=Gvishiani%2C+Alexei+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> (translated from Russian)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnapp2001" class="citation cs2">Knapp, Anthony W. (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QCcW1h835pwC">Representation Theory of Semisimple Groups: An Overview Based on Examples</a>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-09089-4" title="Special:BookSources/978-0-691-09089-4"><bdi>978-0-691-09089-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representation+Theory+of+Semisimple+Groups%3A+An+Overview+Based+on+Examples&rft.pub=Princeton+University+Press&rft.date=2001&rft.isbn=978-0-691-09089-4&rft.aulast=Knapp&rft.aufirst=Anthony+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQCcW1h835pwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmogorovFomin1999" class="citation cs2"><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov, Andrey Nikolaevich</a>; <a href="/wiki/Sergei_Fomin" title="Sergei Fomin">Fomin, Sergei Vasilyevich</a> (1999) [1957], <a rel="nofollow" class="external text" href="http://store.doverpublications.com/0486406830.html">Elements of the Theory of Functions and Functional Analysis</a>, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+Theory+of+Functions+and+Functional+Analysis&rft.pub=Dover&rft.date=1999&rft.aulast=Kolmogorov&rft.aufirst=Andrey+Nikolaevich&rft.au=Fomin%2C+Sergei+Vasilyevich&rft_id=http%3A%2F%2Fstore.doverpublications.com%2F0486406830.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"> (translated from Russian)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLado1971" class="citation cs2">Lado, F. (1971), <a rel="nofollow" class="external text" href="http://www.lib.ncsu.edu/resolver/1840.2/2465">"Numerical Fourier transforms in one, two, and three dimensions for liquid state calculations"</a>, <a href="/wiki/Journal_of_Computational_Physics" title="Journal of Computational Physics">Journal of Computational Physics</a>, 8 (3): 417–433, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971JCoPh...8..417L">1971JCoPh...8..417L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2871%2990021-0">10.1016/0021-9991(71)90021-0</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=Numerical+Fourier+transforms+in+one%2C+two%2C+and+three+dimensions+for+liquid+state+calculations&rft.volume=8&rft.issue=3&rft.pages=417-433&rft.date=1971&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2871%2990021-0&rft_id=info%3Abibcode%2F1971JCoPh...8..417L&rft.aulast=Lado&rft.aufirst=F.&rft_id=http%3A%2F%2Fwww.lib.ncsu.edu%2Fresolver%2F1840.2%2F2465&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller2015" class="citation cs2">Müller, Meinard (2015), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160408083515/https://www.audiolabs-erlangen.de/content/05-fau/professor/00-mueller/04-bookFMP/2015_Mueller_FundamentalsMusicProcessing_Springer_Section2-1_SamplePages.pdf">The Fourier Transform in a Nutshell.</a> (PDF), <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-21945-5">10.1007/978-3-319-21945-5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-21944-8" title="Special:BookSources/978-3-319-21944-8"><bdi>978-3-319-21944-8</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8691186">8691186</a>, archived from <a rel="nofollow" class="external text" href="https://www.audiolabs-erlangen.de/content/05-fau/professor/00-mueller/04-bookFMP/2015_Mueller_FundamentalsMusicProcessing_Springer_Section2-1_SamplePages.pdf">the original</a> (PDF) on 2016-04-08, retrieved 2016-03-28</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fourier+Transform+in+a+Nutshell.&rft.pub=Springer&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8691186%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-319-21945-5&rft.isbn=978-3-319-21944-8&rft.aulast=M%C3%BCller&rft.aufirst=Meinard&rft_id=https%3A%2F%2Fwww.audiolabs-erlangen.de%2Fcontent%2F05-fau%2Fprofessor%2F00-mueller%2F04-bookFMP%2F2015_Mueller_FundamentalsMusicProcessing_Springer_Section2-1_SamplePages.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988">; also available at <a rel="nofollow" class="external text" href="http://www.music-processing.de">Fundamentals of Music Processing</a>, Section 2.1, pages 40–56</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOppenheimSchaferBuck1999" class="citation cs2"><a href="/wiki/Alan_V._Oppenheim" title="Alan V. Oppenheim">Oppenheim, Alan V.</a>; <a href="/wiki/Ronald_W._Schafer" title="Ronald W. Schafer">Schafer, Ronald W.</a>; Buck, John R. (1999), <a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan">Discrete-time signal processing</a> (2nd ed.), Upper Saddle River, N.J.: Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-754920-2" title="Special:BookSources/0-13-754920-2"><bdi>0-13-754920-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete-time+signal+processing&rft.place=Upper+Saddle+River%2C+N.J.&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=0-13-754920-2&rft.aulast=Oppenheim&rft.aufirst=Alan+V.&rft.au=Schafer%2C+Ronald+W.&rft.au=Buck%2C+John+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscretetimesign00alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaleyWiener1934" class="citation cs2"><a href="/wiki/Raymond_Paley" title="Raymond Paley">Paley, R.E.A.C.</a>; <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Wiener, Norbert</a> (1934), Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Providence, Rhode Island: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Transforms+in+the+Complex+Domain&rft.place=Providence%2C+Rhode+Island&rft.series=American+Mathematical+Society+Colloquium+Publications&rft.pub=American+Mathematical+Society&rft.date=1934&rft.aulast=Paley&rft.aufirst=R.E.A.C.&rft.au=Wiener%2C+Norbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPinsky2002" class="citation cs2">Pinsky, Mark (2002), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PyISCgAAQBAJ&q=%22The+Fourier+transform+of+the+measure%22&pg=PA256">Introduction to Fourier Analysis and Wavelets</a>, Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-37660-4" title="Special:BookSources/978-0-534-37660-4"><bdi>978-0-534-37660-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fourier+Analysis+and+Wavelets&rft.pub=Brooks%2FCole&rft.date=2002&rft.isbn=978-0-534-37660-4&rft.aulast=Pinsky&rft.aufirst=Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPyISCgAAQBAJ%26q%3D%2522The%2BFourier%2Btransform%2Bof%2Bthe%2Bmeasure%2522%26pg%3DPA256&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1895" class="citation cs2"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré, Henri</a> (1895), <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k5500702f">Théorie analytique de la propagation de la chaleur</a>, Paris: Carré</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+propagation+de+la+chaleur&rft.place=Paris&rft.pub=Carr%C3%A9&rft.date=1895&rft.aulast=Poincar%C3%A9&rft.aufirst=Henri&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k5500702f&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolyaninManzhirov1998" class="citation cs2">Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-2876-3" title="Special:BookSources/978-0-8493-2876-3"><bdi>978-0-8493-2876-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Integral+Equations&rft.place=Boca+Raton&rft.pub=CRC+Press&rft.date=1998&rft.isbn=978-0-8493-2876-3&rft.aulast=Polyanin&rft.aufirst=A.+D.&rft.au=Manzhirov%2C+A.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressFlanneryTeukolskyVetterling1992" class="citation cs2">Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), Numerical Recipes in C: The Art of Scientific Computing, Second Edition (2nd ed.), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Recipes+in+C%3A+The+Art+of+Scientific+Computing%2C+Second+Edition&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1992&rft.aulast=Press&rft.aufirst=William+H.&rft.au=Flannery%2C+Brian+P.&rft.au=Teukolsky%2C+Saul+A.&rft.au=Vetterling%2C+William+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProakisManolakis1996" class="citation book cs1">Proakis, John G.; Manolakis, Dimitri G. (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/digitalsignalpro00proa">Digital Signal Processing: Principles, Algorithms and Applications</a> (3 ed.). New Jersey: Prentice-Hall International. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996dspp.book.....P">1996dspp.book.....P</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780133942897" title="Special:BookSources/9780133942897"><bdi>9780133942897</bdi></a>. sAcfAQAAIAAJ.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing%3A+Principles%2C+Algorithms+and+Applications&rft.place=New+Jersey&rft.edition=3&rft.pub=Prentice-Hall+International&rft.date=1996&rft_id=info%3Abibcode%2F1996dspp.book.....P&rft.isbn=9780133942897&rft.aulast=Proakis&rft.aufirst=John+G.&rft.au=Manolakis%2C+Dimitri+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdigitalsignalpro00proa&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRahman2011" class="citation cs2">Rahman, Matiur (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=k_rdcKaUdr4C&pg=PA10">Applications of Fourier Transforms to Generalized Functions</a>, WIT Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84564-564-9" title="Special:BookSources/978-1-84564-564-9"><bdi>978-1-84564-564-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Fourier+Transforms+to+Generalized+Functions&rft.pub=WIT+Press&rft.date=2011&rft.isbn=978-1-84564-564-9&rft.aulast=Rahman&rft.aufirst=Matiur&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dk_rdcKaUdr4C%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation cs2">Rudin, Walter (1987), Real and Complex Analysis (3rd ed.), Singapore: McGraw Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-100276-9" title="Special:BookSources/978-0-07-100276-9"><bdi>978-0-07-100276-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis&rft.place=Singapore&rft.edition=3rd&rft.pub=McGraw+Hill&rft.date=1987&rft.isbn=978-0-07-100276-9&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimonenOlkkonen1985" class="citation cs2">Simonen, P.; Olkkonen, H. (1985), "Fast method for computing the Fourier integral transform via Simpson's numerical integration", Journal of Biomedical Engineering, 7 (4): 337–340, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0141-5425%2885%2990067-6">10.1016/0141-5425(85)90067-6</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/4057997">4057997</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Biomedical+Engineering&rft.atitle=Fast+method+for+computing+the+Fourier+integral+transform+via+Simpson%27s+numerical+integration&rft.volume=7&rft.issue=4&rft.pages=337-340&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0141-5425%2885%2990067-6&rft_id=info%3Apmid%2F4057997&rft.aulast=Simonen&rft.aufirst=P.&rft.au=Olkkonen%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith" class="citation web cs1">Smith, Julius O. <a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/~jos/mdft/Positive_Negative_Frequencies.html">"Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition"</a>. ccrma.stanford.edu. Retrieved 2022-12-29. <q>We may think of a real sinusoid as being the sum of a positive-frequency and a negative-frequency complex sinusoid.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ccrma.stanford.edu&rft.atitle=Mathematics+of+the+Discrete+Fourier+Transform+%28DFT%29%2C+with+Audio+Applications+---+Second+Edition&rft.aulast=Smith&rft.aufirst=Julius+O.&rft_id=http%3A%2F%2Fccrma.stanford.edu%2F~jos%2Fmdft%2FPositive_Negative_Frequencies.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2003" class="citation cs2">Stein, Elias; Shakarchi, Rami (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FAOc24bTfGkC&q=%22The+mathematical+thrust+of+the+principle%22&pg=PA158">Fourier Analysis: An introduction</a>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11384-5" title="Special:BookSources/978-0-691-11384-5"><bdi>978-0-691-11384-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis%3A+An+introduction&rft.pub=Princeton+University+Press&rft.date=2003&rft.isbn=978-0-691-11384-5&rft.aulast=Stein&rft.aufirst=Elias&rft.au=Shakarchi%2C+Rami&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFAOc24bTfGkC%26q%3D%2522The%2Bmathematical%2Bthrust%2Bof%2Bthe%2Bprinciple%2522%26pg%3DPA158&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinWeiss1971" class="citation cs2"><a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a>; <a href="/wiki/Guido_Weiss" title="Guido Weiss">Weiss, Guido</a> (1971), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YUCV678MNAIC&q=editions:xbArf-TFDSEC">Introduction to Fourier Analysis on Euclidean Spaces</a>, Princeton, N.J.: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08078-9" title="Special:BookSources/978-0-691-08078-9"><bdi>978-0-691-08078-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fourier+Analysis+on+Euclidean+Spaces&rft.place=Princeton%2C+N.J.&rft.pub=Princeton+University+Press&rft.date=1971&rft.isbn=978-0-691-08078-9&rft.aulast=Stein&rft.aufirst=Elias&rft.au=Weiss%2C+Guido&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYUCV678MNAIC%26q%3Deditions%3AxbArf-TFDSEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaneja2008" class="citation cs2">Taneja, H.C. 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G. (1995), Fourier Series and Optical Transform Techniques in Contemporary Optics, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30357-2" title="Special:BookSources/978-0-471-30357-2"><bdi>978-0-471-30357-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Series+and+Optical+Transform+Techniques+in+Contemporary+Optics&rft.place=New+York&rft.pub=Wiley&rft.date=1995&rft.isbn=978-0-471-30357-2&rft.aulast=Wilson&rft.aufirst=R.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolf1979" class="citation cs2">Wolf, Kurt B. (1979), <a rel="nofollow" class="external text" href="https://www.fis.unam.mx/~bwolf/integraleng.html">Integral Transforms in Science and Engineering</a>, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-0872-1">10.1007/978-1-4757-0872-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-0874-5" title="Special:BookSources/978-1-4757-0874-5"><bdi>978-1-4757-0874-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integral+Transforms+in+Science+and+Engineering&rft.pub=Springer&rft.date=1979&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-0872-1&rft.isbn=978-1-4757-0874-5&rft.aulast=Wolf&rft.aufirst=Kurt+B.&rft_id=https%3A%2F%2Fwww.fis.unam.mx%2F~bwolf%2Fintegraleng.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYosida1968" class="citation cs2"><a href="/wiki/K%C5%8Dsaku_Yosida" title="Kōsaku Yosida">Yosida, K.</a> (1968), Functional Analysis, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58654-8" title="Special:BookSources/978-3-540-58654-8"><bdi>978-3-540-58654-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.pub=Springer&rft.date=1968&rft.isbn=978-3-540-58654-8&rft.aulast=Yosida&rft.aufirst=K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Fourier_transform&action=edit&section=74" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Fourier_transformation" class="extiw" title="commons:Category:Fourier transformation">Fourier transformation</a> at Wikimedia Commons</li> <li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Fourier_transform">Encyclopedia of Mathematics</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FourierTransform.html">"Fourier Transform"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Fourier+Transform&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFourierTransform.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFourier+transform" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.xtal.iqf.csic.es/Cristalografia/parte_05-en.html">Fourier Transform in Crystallography</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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