Transformation de Fourier

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class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés_de_la_transformation_de_Fourier" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriétés_de_la_transformation_de_Fourier"> <div class="vector-toc-text"> 1.4 Propriétés de la transformation de Fourier </div> </a> <ul id="toc-Propriétés_de_la_transformation_de_Fourier-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformation_de_Fourier_inverse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformation_de_Fourier_inverse"> <div class="vector-toc-text"> 1.5 Transformation de Fourier inverse </div> </a> <ul id="toc-Transformation_de_Fourier_inverse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_à_l'espace_ℝn" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extension_à_l'espace_ℝn"> <div class="vector-toc-text"> 1.6 Extension à l'espace ℝn </div> </a> <ul id="toc-Extension_à_l'espace_ℝn-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transformation_de_Fourier_pour_les_fonctions_de_carré_sommable" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Transformation_de_Fourier_pour_les_fonctions_de_carré_sommable"> <div class="vector-toc-text"> 2 Transformation de Fourier pour les fonctions de carré sommable </div> </a> <button aria-controls="toc-Transformation_de_Fourier_pour_les_fonctions_de_carré_sommable-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Transformation de Fourier pour les fonctions de carré sommable </button> <ul id="toc-Transformation_de_Fourier_pour_les_fonctions_de_carré_sommable-sublist" class="vector-toc-list"> <li id="toc-Extension_de_la_transformation_de_L1∩L2_à_L2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extension_de_la_transformation_de_L1∩L2_à_L2"> <div class="vector-toc-text"> 2.1 Extension de la transformation de L1∩L2 à L2 </div> </a> <ul id="toc-Extension_de_la_transformation_de_L1∩L2_à_L2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_transformation_vue_comme_opérateur_de_L2(ℝ)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_transformation_vue_comme_opérateur_de_L2(ℝ)"> <div class="vector-toc-text"> 2.2 La transformation vue comme opérateur de L2(ℝ) </div> </a> <ul id="toc-La_transformation_vue_comme_opérateur_de_L2(ℝ)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lien_avec_le_produit_de_convolution" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lien_avec_le_produit_de_convolution"> <div class="vector-toc-text"> 3 Lien avec le produit de convolution </div> </a> <ul id="toc-Lien_avec_le_produit_de_convolution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principe_d'incertitude" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Principe_d'incertitude"> <div class="vector-toc-text"> 4 Principe d'incertitude </div> </a> <ul id="toc-Principe_d'incertitude-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformation_de_Fourier_sur_l'espace_de_Schwartz" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Transformation_de_Fourier_sur_l'espace_de_Schwartz"> <div class="vector-toc-text"> 5 Transformation de Fourier sur l'espace de Schwartz </div> </a> <ul id="toc-Transformation_de_Fourier_sur_l'espace_de_Schwartz-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformation_de_Fourier_pour_les_distributions_tempérées" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Transformation_de_Fourier_pour_les_distributions_tempérées"> <div class="vector-toc-text"> 6 Transformation de Fourier pour les distributions tempérées </div> </a> <button aria-controls="toc-Transformation_de_Fourier_pour_les_distributions_tempérées-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Transformation de Fourier pour les distributions tempérées </button> <ul id="toc-Transformation_de_Fourier_pour_les_distributions_tempérées-sublist" class="vector-toc-list"> <li id="toc-Compatibilités" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compatibilités"> <div class="vector-toc-text"> 6.1 Compatibilités </div> </a> <ul id="toc-Compatibilités-sublist" class="vector-toc-list"> <li id="toc-Compatibilité_avec_les_espaces_de_fonctions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Compatibilité_avec_les_espaces_de_fonctions"> <div class="vector-toc-text"> 6.1.1 Compatibilité avec les espaces de fonctions </div> </a> <ul id="toc-Compatibilité_avec_les_espaces_de_fonctions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compatibilité_avec_les_espaces_de_suites" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Compatibilité_avec_les_espaces_de_suites"> <div class="vector-toc-text"> 6.1.2 Compatibilité avec les espaces de suites </div> </a> <ul id="toc-Compatibilité_avec_les_espaces_de_suites-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Signaux_discrets_et_signaux_périodiques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signaux_discrets_et_signaux_périodiques"> <div class="vector-toc-text"> 6.2 Signaux discrets et signaux périodiques </div> </a> <ul id="toc-Signaux_discrets_et_signaux_périodiques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Liens_avec_d'autres_transformations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Liens_avec_d'autres_transformations"> <div class="vector-toc-text"> 7 Liens avec d'autres transformations </div> </a> <button aria-controls="toc-Liens_avec_d'autres_transformations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Liens avec d'autres transformations </button> <ul id="toc-Liens_avec_d'autres_transformations-sublist" class="vector-toc-list"> <li id="toc-Lien_avec_les_transformations_de_Laplace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lien_avec_les_transformations_de_Laplace"> <div class="vector-toc-text"> 7.1 Lien avec les transformations de Laplace </div> </a> <ul id="toc-Lien_avec_les_transformations_de_Laplace-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lien_avec_les_séries_de_Fourier" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lien_avec_les_séries_de_Fourier"> <div class="vector-toc-text"> 7.2 Lien avec les séries de Fourier </div> </a> <ul id="toc-Lien_avec_les_séries_de_Fourier-sublist" class="vector-toc-list"> <li id="toc-Parallèle_formel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Parallèle_formel"> <div class="vector-toc-text"> 7.2.1 Parallèle formel </div> </a> <ul id="toc-Parallèle_formel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lien_direct" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lien_direct"> <div class="vector-toc-text"> 7.2.2 Lien direct </div> </a> <ul id="toc-Lien_direct-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autre_interprétation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Autre_interprétation"> <div class="vector-toc-text"> 7.2.3 Autre interprétation </div> </a> <ul id="toc-Autre_interprétation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformée" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transformée"> <div class="vector-toc-text"> 7.2.4 Transformée </div> </a> <ul id="toc-Transformée-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Généralisation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Généralisation"> <div class="vector-toc-text"> 8 Généralisation </div> </a> <ul id="toc-Généralisation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tables_des_principales_transformées_de_Fourier" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Tables_des_principales_transformées_de_Fourier"> <div class="vector-toc-text"> 9 Tables des principales transformées de Fourier </div> </a> <button aria-controls="toc-Tables_des_principales_transformées_de_Fourier-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Tables des principales transformées de Fourier </button> <ul id="toc-Tables_des_principales_transformées_de_Fourier-sublist" class="vector-toc-list"> <li id="toc-Relations_entre_fonctions_à_une_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relations_entre_fonctions_à_une_variable"> <div class="vector-toc-text"> 9.1 Relations entre fonctions à une variable </div> </a> <ul id="toc-Relations_entre_fonctions_à_une_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonctions_de_carré_intégrable_à_une_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fonctions_de_carré_intégrable_à_une_variable"> <div class="vector-toc-text"> 9.2 Fonctions de carré intégrable à une variable </div> </a> <ul id="toc-Fonctions_de_carré_intégrable_à_une_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributions_à_une_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributions_à_une_variable"> <div class="vector-toc-text"> 9.3 Distributions à une variable </div> </a> <ul id="toc-Distributions_à_une_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonctions_de_deux_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fonctions_de_deux_variables"> <div class="vector-toc-text"> 9.4 Fonctions de deux variables </div> </a> <ul id="toc-Fonctions_de_deux_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> 10 Notes et références </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> 11 Voir aussi </div> </a> <button aria-controls="toc-Voir_aussi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Afficher / masquer la sous-section Voir aussi </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> 11.1 Articles connexes </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> 11.2 Bibliographie </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liens_externes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liens_externes"> <div class="vector-toc-text"> 11.3 Liens externes </div> </a> <ul id="toc-Liens_externes-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table des matières" > <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="Basculer la table des matières" > <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" > Basculer la table des matières </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">Transformation de Fourier</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" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 64 langues." > <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 langues </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 – amharique" lang="am" hreflang="am" data-title="የፎሪየር ሽግግር" data-language-autonym="አማርኛ" data-language-local-name="amharique" 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 – arabe" lang="ar" hreflang="ar" data-title="تحويل فورييه" data-language-autonym="العربية" data-language-local-name="arabe" 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 – asturien" lang="ast" hreflang="ast" data-title="Tresformada de Fourier" data-language-autonym="Asturianu" data-language-local-name="asturien" 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 – azerbaïdjanais" lang="az" hreflang="az" data-title="Furye çevrilməsi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Fouriertransformation" title="Fouriertransformation – bavarois" lang="bar" hreflang="bar" data-title="Fouriertransformation" data-language-autonym="Boarisch" data-language-local-name="bavarois" class="interlanguage-link-target">Boarisch</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 – biélorusse" lang="be" hreflang="be" data-title="Пераўтварэнне Фур’е" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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 – bulgare" lang="bg" hreflang="bg" data-title="Преобразование на Фурие" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target">Български</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 – bengali" lang="bn" hreflang="bn" data-title="ফুরিয়ে রূপান্তর" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target">বাংলা</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 – tchèque" lang="cs" hreflang="cs" data-title="Fourierova transformace" data-language-autonym="Čeština" data-language-local-name="tchèque" 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 – danois" lang="da" hreflang="da" data-title="Fouriertransformation" data-language-autonym="Dansk" data-language-local-name="danois" 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 – allemand" lang="de" hreflang="de" data-title="Fourier-Transformation" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Ponimban_Fourier" title="Ponimban Fourier – dusun central" lang="dtp" hreflang="dtp" data-title="Ponimban Fourier" data-language-autonym="Kadazandusun" data-language-local-name="dusun central" class="interlanguage-link-target">Kadazandusun</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 – grec" lang="el" hreflang="el" data-title="Μετασχηματισμός Φουριέ" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform – anglais" lang="en" hreflang="en" data-title="Fourier transform" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Furiera_transformo" title="Furiera transformo – espéranto" lang="eo" hreflang="eo" data-title="Furiera transformo" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target">Esperanto</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 – espagnol" lang="es" hreflang="es" data-title="Transformada de Fourier" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Fourier%27_teisendus" title="Fourier' teisendus – estonien" lang="et" hreflang="et" data-title="Fourier' teisendus" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target">Eesti</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 – persan" lang="fa" hreflang="fa" data-title="تبدیل فوریه" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fourier-muunnos" title="Fourier-muunnos – finnois" lang="fi" hreflang="fi" data-title="Fourier-muunnos" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target">Suomi</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 – galicien" lang="gl" hreflang="gl" data-title="Transformada de Fourier" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target">Galego</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 – hébreu" lang="he" hreflang="he" data-title="התמרת פורייה" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target">עברית</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 – croate" lang="hr" hreflang="hr" data-title="Fourierova transformacija" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target">Hrvatski</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 – hongrois" lang="hu" hreflang="hu" data-title="Fourier-transzformáció" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Transformasi_Fourier" title="Transformasi Fourier – indonésien" lang="id" hreflang="id" data-title="Transformasi Fourier" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" 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 – islandais" lang="is" hreflang="is" data-title="Fourier–vörpun" data-language-autonym="Íslenska" data-language-local-name="islandais" 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 – italien" lang="it" hreflang="it" data-title="Trasformata di Fourier" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target">Italiano</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 – japonais" lang="ja" hreflang="ja" data-title="フーリエ変換" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target">日本語</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-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 – coréen" lang="ko" hreflang="ko" data-title="푸리에 변환" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target">한국어</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 – cachemiri" lang="ks" hreflang="ks" data-title="فوریر ٹرانسفارم" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="cachemiri" class="interlanguage-link-target">कॉशुर / کٲشُر</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 – lituanien" lang="lt" hreflang="lt" data-title="Furjė transformacija" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target">Lietuvių</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 – macédonien" lang="mk" hreflang="mk" data-title="Фуриеова преобразба" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target">Македонски</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 – mongol" lang="mn" hreflang="mn" data-title="Фурье хувиргалт" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target">Монгол</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 – maltais" lang="mt" hreflang="mt" data-title="Trasformata ta' Fourier" data-language-autonym="Malti" data-language-local-name="maltais" class="interlanguage-link-target">Malti</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 – birman" lang="my" hreflang="my" data-title="ဖိုရီယာ ထရန်စဖောင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" 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 – néerlandais" lang="nl" hreflang="nl" data-title="Fouriertransformatie" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fourier-transformasjon" title="Fourier-transformasjon – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Fourier-transformasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fourier-transformasjon" title="Fourier-transformasjon – norvégien bokmål" lang="nb" hreflang="nb" data-title="Fourier-transformasjon" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target">Norsk bokmål</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 – pendjabi" lang="pa" hreflang="pa" data-title="ਫੋਰੀਅਰ ਪਰਿਵਰਤਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendjabi" 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 – polonais" lang="pl" hreflang="pl" data-title="Transformacja Fouriera" data-language-autonym="Polski" data-language-local-name="polonais" 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 – portugais" lang="pt" hreflang="pt" data-title="Transformada de Fourier" data-language-autonym="Português" data-language-local-name="portugais" 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 – roumain" lang="ro" hreflang="ro" data-title="Transformata Fourier" data-language-autonym="Română" data-language-local-name="roumain" 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 – russe" lang="ru" hreflang="ru" data-title="Преобразование Фурье" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target">Русский</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 – slovaque" lang="sk" hreflang="sk" data-title="Fourierova transformácia" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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 – slovène" lang="sl" hreflang="sl" data-title="Fourierova transformacija" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target">Slovenščina</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 – albanais" lang="sq" hreflang="sq" data-title="Transformimi i Furierit" data-language-autonym="Shqip" data-language-local-name="albanais" class="interlanguage-link-target">Shqip</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 – serbe" lang="sr" hreflang="sr" data-title="Фуријеова трансформација" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Transformasi_Fourier" title="Transformasi Fourier – soundanais" lang="su" hreflang="su" data-title="Transformasi Fourier" data-language-autonym="Sunda" data-language-local-name="soundanais" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fouriertransform" title="Fouriertransform – suédois" lang="sv" hreflang="sv" data-title="Fouriertransform" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%82%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AF%87_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81" title="வூரியே மாற்று – tamoul" lang="ta" hreflang="ta" data-title="வூரியே மாற்று" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%9F%E0%B8%B9%E0%B8%A3%E0%B8%B5%E0%B9%80%E0%B8%A2" title="การแปลงฟูรีเย – thaï" lang="th" hreflang="th" data-title="การแปลงฟูรีเย" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fourier_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC" title="Fourier dönüşümü – turc" lang="tr" hreflang="tr" data-title="Fourier dönüşümü" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A4%D1%83%D1%80%D1%8C%D0%B5_%D1%80%D3%99%D0%B2%D0%B5%D1%88%D2%AF%D0%B7%D0%B3%D3%99%D1%80%D1%82%D2%AF%D0%B5" title="Фурье рәвешүзгәртүе – tatar" lang="tt" hreflang="tt" data-title="Фурье рәвешүзгәртүе" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%A4%D1%83%D1%80%27%D1%94" title="Перетворення Фур'є – ukrainien" lang="uk" hreflang="uk" data-title="Перетворення Фур'є" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_%C4%91%E1%BB%95i_Fourier" title="Biến đổi Fourier – vietnamien" lang="vi" hreflang="vi" data-title="Biến đổi Fourier" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换 – wu" lang="wuu" hreflang="wuu" data-title="傅里叶变换" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换 – chinois" lang="zh" hreflang="zh" data-title="傅里叶变换" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target">中文</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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%82%85%E5%88%A9%E8%91%89%E8%AE%8A%E6%8F%9B" title="傅利葉變換 – cantonais" lang="yue" hreflang="yue" data-title="傅利葉變換" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q6520159#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Fourier2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Fourier2.jpg/220px-Fourier2.jpg" decoding="async" width="220" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Fourier2.jpg/330px-Fourier2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Fourier2.jpg/440px-Fourier2.jpg 2x" data-file-width="1619" data-file-height="2048" /></a><figcaption>Portrait de <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>.</figcaption></figure> <div class="infobox_v3 infobox infobox--frwiki noarchive large"><div class="entete" style="background-color:#B33324;color:#FFF"><div>Transformée de Fourier</div></div><div><div class="images" style="padding:2px 0"><a href="/wiki/Fichier:Fourier_transform_time_and_frequency_domains.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Fourier_transform_time_and_frequency_domains.gif/260px-Fourier_transform_time_and_frequency_domains.gif" decoding="async" width="260" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Fourier_transform_time_and_frequency_domains.gif/390px-Fourier_transform_time_and_frequency_domains.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/50/Fourier_transform_time_and_frequency_domains.gif 2x" data-file-width="500" data-file-height="400" /></a></div></div><table><tbody><tr class=""><th scope="row">Type</th><td class=""><div> Concept mathématique (<a href="https://en.wikipedia.org/wiki/Mathematical_concept" class="extiw" title="en:Mathematical concept">en</a>), <a href="/wiki/Op%C3%A9rateur_int%C3%A9gral" title="Opérateur intégral">transformée intégrale</a><a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P31" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr><tr class=""><th scope="row">Nom court</th><td class=""><div> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> FT<a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P1813" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr><tr class=""><th scope="row">Nommé en référence à</th><td class=""><div> <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a><a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P138" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr><tr class=""><th scope="row">Décrit par</th><td class=""><div> ISO 80000-2:2019 (<a href="https://www.wikidata.org/wiki/Q109490582" class="extiw" title="d:Q109490582">d</a>)<a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P1343" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr><tr class=""><th scope="row">Aspect de</th><td class=""><div> <a href="/wiki/Analyse_harmonique_(math%C3%A9matiques)" title="Analyse harmonique (mathématiques)">Analyse de Fourier</a><a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P1269" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr><tr class=""><th scope="row">Formule</th><td class=""><div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {F}}f\right)(\omega )=\int \limits _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \omega t}f(t)\mathrm {d} t}"> <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"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {F}}f\right)(\omega )=\int \limits _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \omega t}f(t)\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356afd7cf6afa142ff2e8ff06d619c8ab46b13cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.687ex; height:8.843ex;" alt="{\displaystyle \left({\mathcal {F}}f\right)(\omega )=\int \limits _{-\infty }^{\infty }\mathrm {e} ^{-\mathrm {i} \omega t}f(t)\mathrm {d} t}" /><a href="https://www.wikidata.org/wiki/Q6520159?uselang=fr#P2534" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></td></tr></tbody></table><a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=0">modifier</a> - <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Transformation_de_Fourier&action=edit&section=0">modifier le code</a> - <a href="https://www.wikidata.org/wiki/Q6520159" class="extiw" title="d:Q6520159">modifier Wikidata</a><a href="/wiki/Mod%C3%A8le:Infobox_M%C3%A9thode_scientifique" title="Documentation du modèle"><img alt="Documentation du modèle" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/12px-Info_Simple.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/18px-Info_Simple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/24px-Info_Simple.svg.png 2x" data-file-width="512" data-file-height="512" /></a></div> En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, plus précisément en <a href="/wiki/Analyse_(math%C3%A9matiques)" title="Analyse (mathématiques)">analyse</a>, la transformation de Fourier est une extension, pour les fonctions non <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">périodiques</a>, du <a href="/wiki/S%C3%A9ries_de_Fourier" class="mw-redirect" title="Séries de Fourier">développement en série de Fourier des fonctions périodiques</a>. La transformation de Fourier associe à toute fonction <a href="/wiki/Int%C3%A9grabilit%C3%A9" title="Intégrabilité">intégrable</a> définie sur ℝ et à valeurs réelles ou complexes, une autre fonction sur ℝ appelée transformée de Fourier dont la <a href="/wiki/Variable_ind%C3%A9pendante" title="Variable indépendante">variable indépendante</a> peut s'interpréter en physique comme la <a href="/wiki/Fr%C3%A9quence" title="Fréquence">fréquence</a> ou la <a href="/wiki/Vitesse_angulaire" title="Vitesse angulaire">pulsation</a>. La transformée de Fourier représente une fonction par la densité spectrale dont elle provient, en tant que moyenne de <a href="/wiki/Fonctions_trigonom%C3%A9triques" class="mw-redirect" title="Fonctions trigonométriques">fonctions trigonométriques</a> de toutes fréquences. La <a href="/wiki/Th%C3%A9orie_de_la_mesure" title="Théorie de la mesure">théorie de la mesure</a> ainsi que la <a href="/wiki/Distribution_(math%C3%A9matiques)" title="Distribution (mathématiques)">théorie des distributions</a> permettent de définir rigoureusement la transformée de Fourier dans toute sa généralité, elle joue un rôle déterminant dans l'<a href="/wiki/Analyse_harmonique_(math%C3%A9matiques)" title="Analyse harmonique (mathématiques)">analyse harmonique</a>. Lorsqu'une fonction représente un <a href="/wiki/Ph%C3%A9nom%C3%A8ne_physique" title="Phénomène physique">phénomène physique</a>, comme l'état du <a href="/wiki/Champ_%C3%A9lectromagn%C3%A9tique" title="Champ électromagnétique">champ électromagnétique</a> ou du <a href="/wiki/Son_(physique)" title="Son (physique)">champ acoustique</a> en un point, on l'appelle <a href="/wiki/Signal_%C3%A9lectrique" title="Signal électrique">signal</a> et sa transformée de Fourier s'appelle son <a href="/wiki/Analyse_spectrale" title="Analyse spectrale">spectre</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Transformation_de_Fourier_pour_les_fonctions_intégrables">Transformation de Fourier pour les fonctions intégrables</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=1" title="Modifier la section : Transformation de Fourier pour les fonctions intégrables" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=1" title="Modifier le code source de la section : Transformation de Fourier pour les fonctions intégrables">modifier le code</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Définition">Définition</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=2" title="Modifier la section : Définition" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=2" title="Modifier le code source de la section : Définition">modifier le code</a>]</div> La transformation de Fourier <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}}}" /> est une opération qui transforme une <a href="/wiki/Fonction_int%C3%A9grable" class="mw-redirect" title="Fonction intégrable">fonction intégrable</a> sur ℝ en une autre fonction, décrivant le <a href="/wiki/Spectre_fr%C3%A9quentiel" title="Spectre fréquentiel">spectre fréquentiel</a> de cette dernière. Si f est une fonction intégrable sur ℝ, sa transformée de Fourier est la fonction <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> <mo stretchy="false">(</mo> <mi>f</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> </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/990f6661e0fa69c8459034caa53eed19294430cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.812ex; height:3.343ex;" alt="{\displaystyle {\mathcal {F}}(f)={\hat {f}}}" /> donnée par la formule<a href="#cite_note-1">[1]</a> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f):\xi \mapsto {\hat {f}}(\xi )=\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} 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> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>ξ</mi> <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> <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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f):\xi \mapsto {\hat {f}}(\xi )=\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0ff1d298904ad8d741f350c1e22647703e2f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.721ex; height:6.176ex;" alt="{\displaystyle {\mathcal {F}}(f):\xi \mapsto {\hat {f}}(\xi )=\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} x}" />.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Conventions_alternatives">Conventions alternatives</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=3" title="Modifier la section : Conventions alternatives" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=3" title="Modifier le code source de la section : Conventions alternatives">modifier le code</a>]</div> Il est possible de choisir une définition alternative pour la transformation de Fourier. Ce choix est une affaire de convention dont les conséquences ne se manifestent (en général) que par des facteurs multiplicatifs constants. Par exemple, certains scientifiques<a href="/wiki/Aide:Pr%C3%A9ciser_un_fait" title="Aide:Préciser un fait">[Lesquels ?]</a> utilisent ainsi<a href="/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" title="Aide:Référence nécessaire">[réf. nécessaire]</a> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f):\nu \mapsto {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} t}"> <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>:</mo> <mi>ν</mi> <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> <mi>ν</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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ν</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f):\nu \mapsto {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e13451f6b8cb5cf5e2903c90bd208a1facf8581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.705ex; height:6.176ex;" alt="{\displaystyle {\mathcal {F}}(f):\nu \mapsto {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} t}" /></dd></dl> avec t en secondes et ν la fréquence (en hertz). Certains utilisent (pour des raisons de symétrie avec la transformation de Fourier inverse) la transformation suivante<a href="#cite_note-2">[2]</a> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f):\omega \mapsto {\hat {f}}(\omega )={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} t}"> <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>:</mo> <mi>ω</mi> <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> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f):\omega \mapsto {\hat {f}}(\omega )={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5534a0acf08d262b276327de1fd737e08bad5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.173ex; height:6.509ex;" alt="{\displaystyle {\mathcal {F}}(f):\omega \mapsto {\hat {f}}(\omega )={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} t}" /></dd></dl> avec t en secondes et ω la pulsation (en radians par seconde). Cette définition n'est cependant pas adaptée au traitement des <a href="/wiki/Produit_de_convolution" title="Produit de convolution">produits de convolution</a> : à cause du facteur <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> </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/df0f39b66b751b232487fb0a3386f3e9088968dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:5.266ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi }}}}" />, on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f*g)\neq {\mathcal {F}}(f)\cdot {\mathcal {F}}(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 stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</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> <mo stretchy="false">(</mo> <mi>f</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> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f*g)\neq {\mathcal {F}}(f)\cdot {\mathcal {F}}(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7b01a5e1e5b29125b4ffc4f2d4e42a4cab67b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.969ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f*g)\neq {\mathcal {F}}(f)\cdot {\mathcal {F}}(g)}" />, à moins d'introduire un tel facteur dans la définition du produit de convolution. L'ensemble de départ est l'ensemble des fonctions intégrables f d'une variable réelle x. L'ensemble d'arrivée est l'ensemble des fonctions d'une variable réelle ξ. Concrètement lorsque cette transformation est utilisée en <a href="/wiki/Traitement_du_signal" title="Traitement du signal">traitement du signal</a>, on notera volontiers t à la place de x et ω ou 2πν à la place de ξ qui seront les variables respectives de temps et de pulsation ou de fréquence. On dira alors que f est dans le domaine temporel, et que <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}}}" /> est dans le domaine fréquentiel. En physique, la transformation de Fourier permet de déterminer le spectre d'un signal. Les phénomènes de <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> donnent une image de l'espace dual du réseau, ils sont une sorte de « machine à transformation de Fourier » naturelle. Pour ces applications, les physiciens définissent en général la transformation directe avec un facteur <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> </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/df0f39b66b751b232487fb0a3386f3e9088968dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:5.266ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi }}}}" /> et la transformation de Fourier inverse avec le même préfacteur. La notation <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 stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</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/24ce6f2d5a1a9e9185c934c0a8950c0d0b1a2714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f)}" /> peut aussi être remplacée par F(ƒ) ou TF(ƒ). Dans cet article, on utilisera exclusivement la première notation. Il est également d'usage dans certaines communautés scientifiques de noter f(x) pour la fonction de départ et f(p) pour sa transformée, faisant ainsi correspondre à x, y, z les variables duales p, q, r. Cette notation est conforme à l'interprétation physique inspirée par la mécanique quantique : dualité entre position et <a href="/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement">quantité de mouvement</a>. Cette notation n'est pas retenue ici. <div class="mw-heading mw-heading3"><h3 id="Extension_de_la_transformation_de_Fourier">Extension de la transformation de Fourier</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=4" title="Modifier la section : Extension de la transformation de Fourier" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=4" title="Modifier le code source de la section : Extension de la transformation de Fourier">modifier le code</a>]</div> Le cadre le plus naturel pour définir les transformations de Fourier est celui des <a href="/wiki/Fonction_int%C3%A9grable" class="mw-redirect" title="Fonction intégrable">fonctions intégrables</a>. Toutefois, de nombreuses opérations (dérivations, transformation de Fourier inverse) ne peuvent être écrites en toute généralité. On doit à <a href="/wiki/Michel_Plancherel" title="Michel Plancherel">Plancherel</a><a href="/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" title="Aide:Référence nécessaire">[réf. nécessaire]</a> l'introduction de la transformation de Fourier pour les fonctions de <a href="/wiki/Carr%C3%A9_sommable" title="Carré sommable">carré sommable</a>, pour lesquelles la formule d'inversion est vraie. Puis la théorie des <a href="/wiki/Distribution_(analyse_math%C3%A9matique)" class="mw-redirect" title="Distribution (analyse mathématique)">distributions</a> de <a href="/wiki/Laurent_Schwartz_(math%C3%A9maticien)" title="Laurent Schwartz (mathématicien)">Schwartz</a>, et plus particulièrement des <a href="/wiki/Distribution_temp%C3%A9r%C3%A9e" title="Distribution tempérée">distributions tempérées</a> permit de trouver un cadre parfaitement adapté. On peut généraliser la définition de la transformation de Fourier à plusieurs variables, et même sur d'autres <a href="/wiki/Groupe_(math%C3%A9matique)" class="mw-redirect" title="Groupe (mathématique)">groupes</a> que le groupe additif ℝn. Ainsi, on peut la définir sur le groupe additif ℝ/ℤ, c'est-à-dire sur les fonctions de période 1 — on retrouve ainsi les <a href="/wiki/S%C3%A9ries_de_Fourier" class="mw-redirect" title="Séries de Fourier">séries de Fourier</a> —, et plus généralement sur des groupes <a href="/wiki/Espace_localement_compact" title="Espace localement compact">localement compacts</a>, pas nécessairement commutatifs, et en particulier sur des groupes finis. Ces définitions font intervenir les <a href="/wiki/Caract%C3%A8re_d%27un_groupe_fini" title="Caractère d'un groupe fini">groupes duaux</a>, ainsi que la <a href="/wiki/Mesure_de_Haar" title="Mesure de Haar">mesure de Haar</a>. <div class="mw-heading mw-heading3"><h3 id="Propriétés_de_la_transformation_de_Fourier">Propriétés de la transformation de Fourier</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=5" title="Modifier la section : Propriétés de la transformation de Fourier" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=5" title="Modifier le code source de la section : Propriétés de la transformation de Fourier">modifier le code</a>]</div> <table class="wikitable center" style="width:80%;"> <tbody><tr> <th> </th> <th scope="col">Fonction </th> <th scope="col">Transformée de Fourier </th></tr> <tr> <th scope="row">Linéarité </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot g_{1}(x)+b\cdot g_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot g_{1}(x)+b\cdot g_{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74837ec06447621ad6c031a6f79a6eed5d60104c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.03ex; height:2.843ex;" alt="{\displaystyle a\cdot g_{1}(x)+b\cdot g_{2}(x)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot {\hat {g}}_{1}(\xi )+b\cdot {\hat {g}}_{2}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</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"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</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"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {\hat {g}}_{1}(\xi )+b\cdot {\hat {g}}_{2}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b9217abde0a3d58cdfc77aacd002c5ff42bb44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.676ex; height:2.843ex;" alt="{\displaystyle a\cdot {\hat {g}}_{1}(\xi )+b\cdot {\hat {g}}_{2}(\xi )}" /> </td></tr> <tr> <th scope="row">Contraction du domaine </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a\cdot x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a\cdot x)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3301a2da5ef3e3ca44ce6aacf4a95bd70e59c5d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.907ex; height:2.843ex;" alt="{\displaystyle f(a\cdot x)\ }" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}\cdot {\hat {f}}(\xi /a)}"> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\cdot {\hat {f}}(\xi /a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0f863712316a0e5333223aad23f9c38e394f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.969ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot {\hat {f}}(\xi /a)}" /> </td></tr> <tr> <th scope="row">Translation temporelle </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x+x_{0})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x+x_{0})\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122bc4d93acacdc02a19ee28ed94d3edb78fc24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.06ex; height:2.843ex;" alt="{\displaystyle g(x+x_{0})\ }" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}(\xi )\cdot \mathrm {e} ^{\mathrm {i} \xi x_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ξ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {g}}(\xi )\cdot \mathrm {e} ^{\mathrm {i} \xi x_{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac4f3b852a70ed2bf2a89e18bedabec5bb9f243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.972ex; height:3.176ex;" alt="{\displaystyle {\hat {g}}(\xi )\cdot \mathrm {e} ^{\mathrm {i} \xi x_{0}}}" /> </td></tr> <tr> <th scope="row">Modulation dans le domaine temporel </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)\cdot \mathrm {e} ^{\mathrm {i} x\xi _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)\cdot \mathrm {e} ^{\mathrm {i} x\xi _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f8f4a9d0d91622cf39a10c8e75591f6dd1a095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.148ex; height:3.176ex;" alt="{\displaystyle g(x)\cdot \mathrm {e} ^{\mathrm {i} x\xi _{0}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}(\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>g</mi> <mo stretchy="false">^</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {g}}(\xi -\xi _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0b070a5d5e3a31e63074eba767ae130c61db44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.984ex; height:2.843ex;" alt="{\displaystyle {\hat {g}}(\xi -\xi _{0})}" /> </td></tr> <tr> <th scope="row">Produit de convolution </th> <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> </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/a436f154de7783ebc43ffa5f4732ec329c9521d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.538ex; height:2.843ex;" alt="{\displaystyle (f*g)(x)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )\cdot {\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>f</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>g</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 )\cdot {\hat {g}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f40f15ac0ccf2ac7c69056681258fd9a27f8e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.288ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi )\cdot {\hat {g}}(\xi )}" /> </td></tr> <tr> <th scope="row">Produit </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\cdot 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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\cdot g)(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d54bea69664ae8347c4bc43326eb170b425e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.022ex; height:2.843ex;" alt="{\displaystyle (f\cdot g)(x)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\xi )}"> <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> <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>∗</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> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42170b568e9f68d171a4c7b21af0211f0a13dab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.105ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\xi )}" /> </td></tr> <tr> <th scope="row">Dérivation dans le domaine temporel </th> <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"> <msup> <mi>f</mi> <mo>′</mo> </msup> <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/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}" /> (voir conditions ci-dessous) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {i}}\xi \cdot {\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 mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {i}}\xi \cdot {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15a0fc4b6796c77354c5fe4daa42416a04b1a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.894ex; height:3.343ex;" alt="{\displaystyle {\rm {i}}\xi \cdot {\hat {f}}(\xi )}" /> </td></tr> <tr> <th scope="row">Dérivation dans le domaine fréquentiel </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dabf37ece4c39a9e4c106f12836f0949517f8896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.426ex; height:2.843ex;" alt="{\displaystyle x\cdot f(x)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} {\hat {f}}'(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</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>′</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} {\hat {f}}'(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74174de46bac46fe7633a8f63869bcba75b10602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.87ex; height:3.676ex;" alt="{\displaystyle \mathrm {i} {\hat {f}}'(\xi )}" /> </td></tr> <tr> <th scope="row" rowspan="5">Symétrie </th> <td>réelle et paire </td> <td>réelle et paire </td></tr> <tr> <td>réelle </td> <td>paire (à symétrie hermitienne) </td></tr> <tr> <td>réelle et impaire </td> <td>imaginaire pure et impaire </td></tr> <tr> <td>imaginaire pure et paire </td> <td>imaginaire pure et paire </td></tr> <tr> <td>imaginaire pure et impaire </td> <td>réelle et impaire </td></tr> <tr> <th scope="row">Forme </th> <td><a href="/wiki/Fonction_gaussienne" title="Fonction gaussienne">gaussienne</a> </td> <td>gaussienne </td></tr></tbody></table> <ul><li>La contraction dans un domaine (temporel, spatial ou fréquentiel) implique une dilatation dans l'autre. Un exemple concret de ce phénomène peut être observé par exemple sur un <a href="/wiki/Tourne-disque" class="mw-redirect" title="Tourne-disque">tourne-disque</a>. La lecture d'un 33 tours à 45 tours par minute implique une augmentation de la fréquence du signal audio (a > 1), on contracte le signal audio dans le domaine temporel ce qui le dilate dans le <a href="/wiki/Domaine_fr%C3%A9quentiel" title="Domaine fréquentiel">domaine fréquentiel</a>.</li> <li>Si la fonction f est à support borné (<abbr class="abbr nowrap" title="c’est-à-dire">c.-à-d.</abbr> si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x_{0}\in \mathbb {R} ,\forall |x|>x_{0},f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <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> <mo>,</mo> <mi mathvariant="normal">∀</mi> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>f</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 \exists x_{0}\in \mathbb {R} ,\forall |x|>x_{0},f(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257f3507754b006eb53a59eb3a8741a6124264bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.34ex; height:2.843ex;" alt="{\displaystyle \exists x_{0}\in \mathbb {R} ,\forall |x|>x_{0},f(x)=0}" />) alors <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}}}" /> est à support infini. Inversement, si le support spectral de la fonction <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}}}" /> est borné alors f est à support non borné.</li> <li>Si f est une fonction non nulle sur un intervalle borné alors <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}}}" /> est une fonction non nulle sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" /> et inversement, si <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}}}" /> est non nulle sur un intervalle borné alors f est une fonction non nulle sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />.</li> <li>La transformée de Fourier de f est une <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">fonction continue</a>, de limite nulle à l'infini (<a href="/wiki/Th%C3%A9or%C3%A8me_de_Riemann-Lebesgue" title="Théorème de Riemann-Lebesgue">théorème de Riemann-Lebesgue</a>), notamment bornée par</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\hat {f}}\|_{\infty }\leq \|f\|_{1}}"> <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>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≤</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\hat {f}}\|_{\infty }\leq \|f\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f9021eafa3057042ae31aab9136836878977e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.656ex; height:3.343ex;" alt="{\displaystyle \|{\hat {f}}\|_{\infty }\leq \|f\|_{1}}" />.</dd></dl> <ul><li>Par <a href="/wiki/Int%C3%A9gration_par_changement_de_variable" title="Intégration par changement de variable">changement de variable</a> on trouve des formules intéressantes lorsqu'on effectue une translation, dilatation du graphe de f.</li> <li>Supposons que la fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:x\mapsto -\mathrm {i} xf(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:x\mapsto -\mathrm {i} xf(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b67d546547dde057323d9ab0ec4cd6246a20b6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.199ex; height:2.843ex;" alt="{\displaystyle g:x\mapsto -\mathrm {i} xf(x)}" /> soit intégrable ; alors on peut dériver la formule de définition <a href="/wiki/D%C3%A9rivation_sous_int%C3%A9grale" class="mw-redirect" title="Dérivation sous intégrale">sous le signe d'intégration</a>. On constate alors que la dérivée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> </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/e5b62fec49bd9fba9a1d9fc6e4379d793187cfe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:3.509ex;" alt="{\displaystyle {\hat {f}}'}" /> est la transformée de Fourier de g.</li> <li>Si f est localement <a href="/wiki/Absolue_continuit%C3%A9" title="Absolue continuité">absolument continue</a> (<abbr class="abbr nowrap" title="c’est-à-dire">c.-à-d.</abbr> dérivable presque partout et égale à « l'intégrale de sa dérivée ») et si f et f ' sont intégrables, alors<a href="#cite_note-3">[3]</a> la transformée de Fourier de la dérivée de f est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f'}}(\xi )=\mathrm {i} \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> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f'}}(\xi )=\mathrm {i} \xi {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/016ffe40f00f1a1b92fb18879a760b8f321ba422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:14.487ex; height:3.676ex;" alt="{\displaystyle {\widehat {f'}}(\xi )=\mathrm {i} \xi {\hat {f}}(\xi )}" />.</li></ul> On peut résumer les deux dernières propriétés : notons D l'opération <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df={\frac {1}{\mathrm {i} }}f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Df={\frac {1}{\mathrm {i} }}f'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05919db8b6c08c8c93768698d5c2b79d37ca6e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.305ex; height:5.176ex;" alt="{\displaystyle Df={\frac {1}{\mathrm {i} }}f'}" /></dd></dl> et M la multiplication par l'argument : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Mf)(x)=xf(x),\quad (M{\hat {f}})(\xi )=\xi {\hat {f}}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>M</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> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Mf)(x)=xf(x),\quad (M{\hat {f}})(\xi )=\xi {\hat {f}}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9665f043bc6c6dd55dcd99ab66ff6cdb13f42244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.328ex; height:3.343ex;" alt="{\displaystyle (Mf)(x)=xf(x),\quad (M{\hat {f}})(\xi )=\xi {\hat {f}}(\xi )}" />.</dd></dl> Alors, si f satisfait des conditions fonctionnelles convenables, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {Df}}=+M{\hat {f}}}"> <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>f</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>+</mo> <mi>M</mi> <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 {\widehat {Df}}=+M{\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e3b66801397889c313440441c26d423891c99a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.493ex; height:3.343ex;" alt="{\displaystyle {\widehat {Df}}=+M{\hat {f}}}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {Mf}}=-D{\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>f</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>D</mi> <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 {\widehat {Mf}}=-D{\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48bf7b2f36b4687839ae264d46d505bfc88d048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.251ex; height:3.343ex;" alt="{\displaystyle {\widehat {Mf}}=-D{\hat {f}}}" />. On s'affranchira de ces conditions fonctionnelles en élargissant la classe des objets sur lesquels opère la transformation de Fourier. C'est une des motivations de la définition des <a href="/wiki/Distribution_(math%C3%A9matiques)" title="Distribution (mathématiques)">distributions</a>. <div class="mw-heading mw-heading3"><h3 id="Transformation_de_Fourier_inverse">Transformation de Fourier inverse</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=6" title="Modifier la section : Transformation de Fourier inverse" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=6" title="Modifier le code source de la section : Transformation de Fourier inverse">modifier le code</a>]</div> Si la transformée de Fourier de f, notée <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}}}" />, est elle-même une <a href="/wiki/Fonction_int%C3%A9grable" class="mw-redirect" title="Fonction intégrable">fonction intégrable</a>, la formule dite de <a href="/w/index.php?title=Transformation_de_Fourier_inverse&action=edit&redlink=1" class="new" title="Transformation de Fourier inverse (page inexistante)">transformation de Fourier inverse</a>, opération notée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{-1}}"> <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"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798153399d91ed4f7c88fa012bd0fabe708c4de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.336ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}^{-1}}" />, et appliquée à <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}}}" />, permet (sous conditions appropriées) de retrouver f à partir des données fréquentielles : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\mathcal {F}}^{-1}({\hat {f}})(x)={1 \over 2\pi }\,\int _{-\infty }^{+\infty }{\hat {f}}(\xi )\,\mathrm {e} ^{+{\rm {i}}\xi x}\,\mathrm {d} \xi \qquad \Leftrightarrow \qquad {\hat {f}}(\xi )\ =\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} 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> <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 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 stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <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> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ξ</mi> <mspace width="2em"></mspace> <mo stretchy="false">⇔</mo> <mspace width="2em"></mspace> <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> <mtext> </mtext> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\mathcal {F}}^{-1}({\hat {f}})(x)={1 \over 2\pi }\,\int _{-\infty }^{+\infty }{\hat {f}}(\xi )\,\mathrm {e} ^{+{\rm {i}}\xi x}\,\mathrm {d} \xi \qquad \Leftrightarrow \qquad {\hat {f}}(\xi )\ =\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ffe413c863b5b9a23257f8337f0ac56753b590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:83.623ex; height:6.176ex;" alt="{\displaystyle f(x)={\mathcal {F}}^{-1}({\hat {f}})(x)={1 \over 2\pi }\,\int _{-\infty }^{+\infty }{\hat {f}}(\xi )\,\mathrm {e} ^{+{\rm {i}}\xi x}\,\mathrm {d} \xi \qquad \Leftrightarrow \qquad {\hat {f}}(\xi )\ =\int _{-\infty }^{+\infty }f(x)\,\mathrm {e} ^{-{\rm {i}}\xi x}\,\mathrm {d} x}" />.</dd></dl> Cette opération de transformation de Fourier inverse a des propriétés analogues à la transformation directe, puisque seuls changent le coefficient multiplicatif et le –i devenu i. Dans le cas des définitions alternatives, la transformation de Fourier inverse devient : <dl><dd>Définition en fréquence : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\int _{-\infty }^{+\infty }{\hat {f}}(\nu )\,\mathrm {e} ^{+{\rm {i}}2\pi \nu t}\,\mathrm {d} \nu \qquad \Leftrightarrow \qquad {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} 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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <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>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ν</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ν</mi> <mspace width="2em"></mspace> <mo stretchy="false">⇔</mo> <mspace width="2em"></mspace> <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>ν</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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ν</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\int _{-\infty }^{+\infty }{\hat {f}}(\nu )\,\mathrm {e} ^{+{\rm {i}}2\pi \nu t}\,\mathrm {d} \nu \qquad \Leftrightarrow \qquad {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee82968eeb76de1720e4d4e57aa04ab83f25ee9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.112ex; height:6.176ex;" alt="{\displaystyle f(t)=\int _{-\infty }^{+\infty }{\hat {f}}(\nu )\,\mathrm {e} ^{+{\rm {i}}2\pi \nu t}\,\mathrm {d} \nu \qquad \Leftrightarrow \qquad {\hat {f}}(\nu )=\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}2\pi \nu t}\,\mathrm {d} t}" />.</dd></dl> <dl><dd>Définition en pulsation : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\hat {f}}(\omega )\,\mathrm {e} ^{+{\rm {i}}\omega t}\,\mathrm {d} \omega \quad \Leftrightarrow \quad {\hat {f}}(\omega )\ ={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} 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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <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>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> <mspace width="1em"></mspace> <mo stretchy="false">⇔</mo> <mspace width="1em"></mspace> <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> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\hat {f}}(\omega )\,\mathrm {e} ^{+{\rm {i}}\omega t}\,\mathrm {d} \omega \quad \Leftrightarrow \quad {\hat {f}}(\omega )\ ={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bb89e3431ebb8284cbc933c09c28eff91b0fe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:71.77ex; height:6.509ex;" alt="{\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\hat {f}}(\omega )\,\mathrm {e} ^{+{\rm {i}}\omega t}\,\mathrm {d} \omega \quad \Leftrightarrow \quad {\hat {f}}(\omega )\ ={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }f(t)\,\mathrm {e} ^{-{\rm {i}}\omega t}\,\mathrm {d} t}" />.</dd></dl> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;"><a href="/wiki/Preuve#Différents_types_de_preuves" title="Preuve">Preuve</a> par la <a href="/wiki/Formule_sommatoire_de_Poisson" title="Formule sommatoire de Poisson">formule sommatoire de Poisson</a></div><div class="NavContent" style="padding-bottom:0.4em"> Soit h une fonction complexe définie sur ℝ et deux fois continûment différentiable. On suppose que h vérifie l'estimation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |h(x)|\leq {\frac {C}{1+x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |h(x)|\leq {\frac {C}{1+x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64dbb638f03ad92004298271a164e68d98710e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.093ex; height:5.843ex;" alt="{\displaystyle |h(x)|\leq {\frac {C}{1+x^{2}}}}" /></dd></dl> et que les deux premières dérivées de h sont intégrables sur ℝ. Alors la transformée de Fourier de h vérifie une estimation analogue <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\hat {h}}(\xi )|\leq {\frac {C}{1+\xi ^{2}}}}"> <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"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <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 |{\hat {h}}(\xi )|\leq {\frac {C}{1+\xi ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed98479691e180f9c752761d5b6f254e701b1802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.497ex; height:6.009ex;" alt="{\displaystyle |{\hat {h}}(\xi )|\leq {\frac {C}{1+\xi ^{2}}}}" />.</dd></dl> Soit y un nombre réel qui, pour le moment, est simplement un paramètre, et notons : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=h(x)\mathrm {e} ^{-\mathrm {i} yx}}"> <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>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>y</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=h(x)\mathrm {e} ^{-\mathrm {i} yx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829a2bfff39175f95974302ec900b71b3ab28ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.752ex; height:3.176ex;" alt="{\displaystyle f(x)=h(x)\mathrm {e} ^{-\mathrm {i} yx}}" />.</dd></dl> On vérifie que f a les mêmes propriétés fonctionnelles que h. Par conséquent, on peut appliquer la <a href="/wiki/Formule_sommatoire_de_Poisson" title="Formule sommatoire de Poisson">formule sommatoire de Poisson</a> à f, avec la période 2π : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in \mathbb {Z} }f(x+2\pi n)={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {f}}(k)\mathrm {e} ^{{\rm {i}}kx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }f(x+2\pi n)={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {f}}(k)\mathrm {e} ^{{\rm {i}}kx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7aa82dacfadabf79d2de5996f32a036d94aa33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.686ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }f(x+2\pi n)={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {f}}(k)\mathrm {e} ^{{\rm {i}}kx}}" />.</dd></dl></dd></dl> Mais le calcul de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(k)}"> <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>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dfea487a49c5d594b09031c15640c717d4d2072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(k)}" /> donne : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(k)=\int _{\mathbb {R} }h(x)\mathrm {e} ^{-{\rm {i}}(y+k)x}\,\mathrm {d} x={\hat {h}}(y+k)}"> <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>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <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>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(k)=\int _{\mathbb {R} }h(x)\mathrm {e} ^{-{\rm {i}}(y+k)x}\,\mathrm {d} x={\hat {h}}(y+k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaba6733b38ba7150e736a45c10ac19d12b0e63c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.03ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}(k)=\int _{\mathbb {R} }h(x)\mathrm {e} ^{-{\rm {i}}(y+k)x}\,\mathrm {d} x={\hat {h}}(y+k)}" />.</dd></dl> On peut donc réécrire la formule sommatoire de Poisson en termes de h, et il vient : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-{\rm {i}}(x+2\pi n)y}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}kx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <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>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-{\rm {i}}(x+2\pi n)y}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}kx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc07a22fd44d91ad1dd579bf363b7edf72fdd0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.448ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-{\rm {i}}(x+2\pi n)y}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}kx}}" />.</dd></dl> On multiplie les deux membres de cette identité par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{\mathrm {i} xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} xy}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9f010d833421f0cf54f441ddcc2ef8f0e41410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.479ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} xy}}" /> : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-2{\rm {i}}\pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}(k+y)x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>n</mi> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <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>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-2{\rm {i}}\pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}(k+y)x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a884bf845e40c4268c5e5fd5084ca3113517fe8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.325ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }h(x+2\pi n)\mathrm {e} ^{-2{\rm {i}}\pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k)\mathrm {e} ^{{\rm {i}}(k+y)x}}" />.</dd></dl></dd></dl> On remarque que les séries apparaissant de part et d'autre sont normalement convergentes pour la norme du maximum. On va donc pouvoir échanger la sommation et l'intégration par rapport à y sur l'intervalle [0 ; 1]. À gauche, l'intégration par rapport à y ne laisse subsister qu'un seul terme, celui correspondant à n = 0. À droite, on intègre par rapport à y et l'on effectue dans chaque intégrale le changement de variable y + k = ξ. On obtient ainsi la formule <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }{\hat {h}}(\xi )\mathrm {e} ^{{\rm {i}}x\xi }\,\mathrm {d} \xi }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }{\hat {h}}(\xi )\mathrm {e} ^{{\rm {i}}x\xi }\,\mathrm {d} \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d282d9be10a6a256b2caa6af33c571dbeee1bac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.671ex; height:5.676ex;" alt="{\displaystyle h(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }{\hat {h}}(\xi )\mathrm {e} ^{{\rm {i}}x\xi }\,\mathrm {d} \xi }" />.</dd></dl></dd></dl> On passe au cas général de la formule d'inversion de Fourier pour une fonction f intégrable ainsi que sa transformée de Fourier par une méthode de <a href="/wiki/Partie_dense" title="Partie dense">densité</a>. On approche f par une suite de fonctions fp vérifiant les hypothèses fonctionnelles de la présente démonstration. On doit bien sûr supposer que les fp et leurs transformées de Fourier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{p}}"> <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>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd02efc4a0a086e89f480bc74d7ad9a2c9f0452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.758ex; height:3.676ex;" alt="{\displaystyle {\hat {f}}_{p}}" /> convergent vers leurs limites respectives f et <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}}}" /> en norme L1(ℝ). On peut construire de telles approximations en tronquant f, c'est-à-dire en le remplaçant par 0 en dehors de l'intervalle [–p, p], et en le régularisant par convolution. Si ϕ est une fonction deux fois continûment différentiable, d'intégrale 1, et à support borné, on pose ϕp(x) = p ϕ(px) et l'on convole la fonction tronquée f|[–p,p] par ϕp. C'est une idée raisonnable d'utiliser ici le même paramètre p. </div><div class="clear" style="clear:both;"></div> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Preuve par l'<a href="/wiki/Analyse_non_standard" title="Analyse non standard">analyse non standard</a></div><div class="NavContent" style="padding-bottom:0.4em"> Soit f une fonction de classe <a href="/wiki/Fonction_C%E2%88%9E_%C3%A0_support_compact" title="Fonction C∞ à support compact">C∞ à support compact</a>. Par le <a href="/wiki/Analyse_non_standard#Axiome_de_transfert" title="Analyse non standard">principe de transfert</a>, on peut se contenter d'étudier le cas d'une fonction <a href="/wiki/Analyse_non_standard" title="Analyse non standard">standard</a>. Dans ce cas, il existe un <a href="/wiki/Analyse_non_standard#Les_réels" title="Analyse non standard">réel infiniment grand</a> T tel que pour tout réel |x| > T, f(x) = 0. Introduisons une <a href="/wiki/Base_hilbertienne" class="mw-redirect" title="Base hilbertienne">base hilbertienne</a> de L2([–T, T]) donnée par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{n}:x\mapsto {\rm {e}}^{\mathrm {i} n\pi x/T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{n}:x\mapsto {\rm {e}}^{\mathrm {i} n\pi x/T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f81a03a6ec0cb6b142b26f23c4c9b108df3ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.752ex; height:3.176ex;" alt="{\displaystyle e_{n}:x\mapsto {\rm {e}}^{\mathrm {i} n\pi x/T}}" /></dd></dl> (un calcul immédiat montre qu'elle est bien orthonormée, et <a href="/wiki/Th%C3%A9or%C3%A8me_de_Riesz-Fischer" title="Théorème de Riesz-Fischer">le fait qu'elle soit totale</a> se déduit de la densité des fonctions continues et de leur <a href="/wiki/Polyn%C3%B4me_trigonom%C3%A9trique#Théorème_de_Stone-Weierstrass" title="Polynôme trigonométrique">approximation uniforme par des polynômes trigonométriques</a>). Par le lemme de Parseval, on est en mesure d'écrire : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{n\in \mathbb {Z} }c_{n}e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{n\in \mathbb {Z} }c_{n}e_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2838c38a0a38fe13995139e6b188f7f2a892967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.646ex; height:5.676ex;" alt="{\displaystyle f=\sum _{n\in \mathbb {Z} }c_{n}e_{n}}" /> où <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={\frac {1}{2T}}\int _{-T}^{T}f(x){\rm {e}}^{-\mathrm {i} n\pi x/T}{\rm {d}}x={\frac {1}{2T}}{\widehat {f}}\left({\frac {n\pi }{T}}\right)}"> <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> <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>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>T</mi> </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> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {1}{2T}}\int _{-T}^{T}f(x){\rm {e}}^{-\mathrm {i} n\pi x/T}{\rm {d}}x={\frac {1}{2T}}{\widehat {f}}\left({\frac {n\pi }{T}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0db736bd3e74cb25a2649c03d219adc3a590cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.811ex; height:6.343ex;" alt="{\displaystyle c_{n}={\frac {1}{2T}}\int _{-T}^{T}f(x){\rm {e}}^{-\mathrm {i} n\pi x/T}{\rm {d}}x={\frac {1}{2T}}{\widehat {f}}\left({\frac {n\pi }{T}}\right)}" /></dd></dl> Plus explicitement, pour x standard : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{2\pi }}\sum _{n\in \mathbb {Z} }{\widehat {f}}\left({\frac {n\pi }{T}}\right){\rm {e}}^{\mathrm {i} n\pi x/T}{\frac {\pi }{T}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f}}(w){\rm {e}}^{{\rm {i}}wx}{\rm {d}}w}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <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> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>n</mi> <mi>π</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>T</mi> </mfrac> </mrow> <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> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>w</mi> <mi>x</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{2\pi }}\sum _{n\in \mathbb {Z} }{\widehat {f}}\left({\frac {n\pi }{T}}\right){\rm {e}}^{\mathrm {i} n\pi x/T}{\frac {\pi }{T}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f}}(w){\rm {e}}^{{\rm {i}}wx}{\rm {d}}w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a86e36be1dec93703b1bd7be0c8c6444245473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.591ex; height:6.676ex;" alt="{\displaystyle f(x)={\frac {1}{2\pi }}\sum _{n\in \mathbb {Z} }{\widehat {f}}\left({\frac {n\pi }{T}}\right){\rm {e}}^{\mathrm {i} n\pi x/T}{\frac {\pi }{T}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\widehat {f}}(w){\rm {e}}^{{\rm {i}}wx}{\rm {d}}w}" />.</dd></dl> La dernière égalité vient de ce que le membre de gauche est standard, que la somme de Riemann s'effectue sur une partition de longueur infiniment petite (π/T), et donc que le membre de droite est la partie standard du membre intermédiaire. L'égalité recherchée est donc vraie pour toutes les fonctions standard de classe C∞ à support compact et tout x standard. Par le principe de transfert, elle est aussi vérifiée pour toutes les fonctions C∞ à support compact et tout x, puis par densité des fonctions C∞ à support compact dans l'espace des fonctions intégrables, pour toutes les fonctions intégrables dont la transformée est intégrable et pour presque tout x. </div><div class="clear" style="clear:both;"></div> </div> <div class="mw-heading mw-heading3"><h3 id="Extension_à_l'espace_ℝn">Extension à l'espace ℝn</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=7" title="Modifier la section : Extension à l'espace ℝn" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=7" title="Modifier le code source de la section : Extension à l'espace ℝn">modifier le code</a>]</div> Notons x∙ξ le <a href="/wiki/Produit_scalaire_canonique" title="Produit scalaire canonique">produit scalaire canonique</a> dans ℝn :<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot \xi =\sum _{j=1}^{n}x_{j}\xi _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot \xi =\sum _{j=1}^{n}x_{j}\xi _{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d6dd89a474ee1a0e010afe1e59cb819e24239d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.047ex; height:7.176ex;" alt="{\displaystyle x\cdot \xi =\sum _{j=1}^{n}x_{j}\xi _{j}}" />.</center> Si f est une <a href="/wiki/Fonction_int%C3%A9grable" class="mw-redirect" title="Fonction intégrable">fonction intégrable</a> sur ℝn, sa transformée de Fourier est donnée par la formule : <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)~{\rm {e}}^{-{\rm {i}}x\cdot \xi }~\mathrm {d} 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> <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> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)~{\rm {e}}^{-{\rm {i}}x\cdot \xi }~\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8284251bb9e63701e34e091e6c24b61866d329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.028ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)~{\rm {e}}^{-{\rm {i}}x\cdot \xi }~\mathrm {d} x}" />.</center> Si A est une isométrie linéaire directe, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f\circ A}}={\hat {f}}\circ A}"> <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>A</mi> </mrow> <mo>^</mo> </mover> </mrow> </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> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f\circ A}}={\hat {f}}\circ A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81273e0adb18672658cd30f5019b2c6a62cb564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.952ex; height:3.509ex;" alt="{\displaystyle {\widehat {f\circ A}}={\hat {f}}\circ A}" />. Il en résulte que la transformée de Fourier d'une fonction radiale est radiale. <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Expression de la transformée de Fourier dans ℝn d'une fonction radiale</div><div class="NavContent" style="padding-bottom:0.4em"> Par définition : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(t_{1},t_{2},\dots ,t_{n})=\int _{\mathbb {R} ^{n}}f(x_{1},x_{2},\dots ,x_{n})~{\rm {e}}^{{\rm {2i\pi }}(x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n})}~\mathrm {d} x_{1}\mathrm {d} x_{2}\dots \mathrm {d} x_{n}}"> <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> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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> <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> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</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>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(t_{1},t_{2},\dots ,t_{n})=\int _{\mathbb {R} ^{n}}f(x_{1},x_{2},\dots ,x_{n})~{\rm {e}}^{{\rm {2i\pi }}(x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n})}~\mathrm {d} x_{1}\mathrm {d} x_{2}\dots \mathrm {d} x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ae964b52745745d938e1164387b3195942b880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:75.581ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}(t_{1},t_{2},\dots ,t_{n})=\int _{\mathbb {R} ^{n}}f(x_{1},x_{2},\dots ,x_{n})~{\rm {e}}^{{\rm {2i\pi }}(x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n})}~\mathrm {d} x_{1}\mathrm {d} x_{2}\dots \mathrm {d} x_{n}}" />.</dd></dl> Si l'on se place dans le cas où f est radiale (ou à symétrie sphérique) alors f ne dépend des variables x1, ..., xn que par l'intermédiaire de la variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\sqrt {x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e015c01343dd5393802683acdbffabd6caf1138c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.185ex; height:4.843ex;" alt="{\displaystyle \rho ={\sqrt {x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}}}}" />. On montre alors que <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}}}" /> ne dépend des variables t1, ..., tn que par l'intermédiaire de la variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\sqrt {t_{1}^{2}+t_{2}^{2}+\dots +t_{n}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\sqrt {t_{1}^{2}+t_{2}^{2}+\dots +t_{n}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e08716a8aa53d1e798a7bb8364d0e5144ea43d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.714ex; height:4.843ex;" alt="{\displaystyle \tau ={\sqrt {t_{1}^{2}+t_{2}^{2}+\dots +t_{n}^{2}}}}" />. <dl><dd>Soit f(x1, ... , xn) = g(ρ).</dd></dl> En notant les vecteurs : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\rho }}=\left({\begin{array}{*{20}{c}}{x_{1}}\\{x_{2}}\\\vdots \\{x_{n}}\end{array}}\right){\text{ et }}{\vec {\tau }}=\left({\begin{array}{*{20}{c}}{t_{1}}\\{t_{2}}\\\vdots \\{t_{n}}\end{array}}\right)\Rightarrow {\vec {\rho }}\cdot {\vec {\tau }}=\rho \tau \cos \theta =x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>ρ</mi> <mi>τ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>t</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>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\rho }}=\left({\begin{array}{*{20}{c}}{x_{1}}\\{x_{2}}\\\vdots \\{x_{n}}\end{array}}\right){\text{ et }}{\vec {\tau }}=\left({\begin{array}{*{20}{c}}{t_{1}}\\{t_{2}}\\\vdots \\{t_{n}}\end{array}}\right)\Rightarrow {\vec {\rho }}\cdot {\vec {\tau }}=\rho \tau \cos \theta =x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0349292324839750e6cb0e8269a8154d0ee8bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:72.551ex; height:13.843ex;" alt="{\displaystyle {\vec {\rho }}=\left({\begin{array}{*{20}{c}}{x_{1}}\\{x_{2}}\\\vdots \\{x_{n}}\end{array}}\right){\text{ et }}{\vec {\tau }}=\left({\begin{array}{*{20}{c}}{t_{1}}\\{t_{2}}\\\vdots \\{t_{n}}\end{array}}\right)\Rightarrow {\vec {\rho }}\cdot {\vec {\tau }}=\rho \tau \cos \theta =x_{1}t_{1}+x_{2}t_{2}+\dots +x_{n}t_{n}}" /> <dl><dd>En passant des coordonnées cartésiennes aux coordonnées polaires dans ℝn :</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}{\vec {\rho }}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </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>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </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> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}{\vec {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a344f528d434f66000a83291f86fbf7a92801a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.288ex; height:5.676ex;" alt="{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}{\vec {\rho }}}" /></dd></dl> Considérons la rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {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">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /> telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\tau }}'={\mathcal {R}}({\vec {\tau }})\Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}'\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <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>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </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> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}'={\mathcal {R}}({\vec {\tau }})\Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}'\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c797bc27df22a55b11df0decdbcb6c4652e5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.543ex; height:5.676ex;" alt="{\displaystyle {\vec {\tau }}'={\mathcal {R}}({\vec {\tau }})\Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\vec {\tau }}'\cdot {\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\vec {\rho }}}~\mathrm {d} ^{n}{\vec {\rho }}}" /> <dl><dd>On ne change pas la valeur de l'intégrale si on remplace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3a6c5c3aafaa0533737546f982ec261ddc8ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.341ex; height:2.843ex;" alt="{\displaystyle {\vec {\rho }}}" /> par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}({\vec {\rho }})}"> <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">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}({\vec {\rho }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3ae2bbf052c37962e1e2b17029a51ab5386a9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.121ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}({\vec {\rho }})}" /> du fait que g est radiale.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\mathcal {R}}({\vec {\rho }})}~\mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})}"> <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"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <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>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\mathcal {R}}({\vec {\rho }})}~\mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48084c82422b1c24a871e9cd9f12c938e82c8473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.377ex; height:5.676ex;" alt="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}{\mathcal {R}}({\vec {\tau }}){\mathcal {R}}({\vec {\rho }})}~\mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})}" /></dd> <dd>Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}({\vec {\tau }})\cdot {\mathcal {R}}({\vec {\rho }})={\vec {\tau }}\cdot {\vec {\rho }}=\tau \rho \cos \theta }"> <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">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}({\vec {\tau }})\cdot {\mathcal {R}}({\vec {\rho }})={\vec {\tau }}\cdot {\vec {\rho }}=\tau \rho \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3285593ce767d2f522f1de61e1840d346ac70c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.844ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}({\vec {\tau }})\cdot {\mathcal {R}}({\vec {\rho }})={\vec {\tau }}\cdot {\vec {\rho }}=\tau \rho \cos \theta }" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})=\mathrm {d} ^{n}{\vec {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})=\mathrm {d} ^{n}{\vec {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f09a9187694d353fa9b26e67f02119bee97714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.583ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} ^{n}{\mathcal {R}}({\vec {\rho }})=\mathrm {d} ^{n}{\vec {\rho }}}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}\rho ={\hat {f}}({\vec {\tau }})}"> <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"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <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>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>ρ</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}\rho ={\hat {f}}({\vec {\tau }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f21e271fb174833cc64b86e23e49fca097df4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.53ex; height:5.676ex;" alt="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }}')=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }~\mathrm {d} ^{n}\rho ={\hat {f}}({\vec {\tau }})}" /></dd></dl> La transformée de Fourier d'une fonction radiale est donc aussi une fonction radiale (qui ne dépend que de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel {\vec {\tau }}\parallel }"> <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"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">∥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \parallel {\vec {\tau }}\parallel }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc41ee8c6b3357a90a1e851b1afc1a1bf51ff711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.658ex; height:2.843ex;" alt="{\displaystyle \parallel {\vec {\tau }}\parallel }" />). <dl><dd>On rappelle la correspondance entre coordonnées sphériques et coordonnées polaires dans ℝn, coordonnées aussi appelées « <a href="/wiki/Coordonn%C3%A9es_hypersph%C3%A9riques" class="mw-redirect" title="Coordonnées hypersphériques">Coordonnées hypersphériques</a> ».</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=\rho \cos(\varphi _{n-i+1})\prod _{k=1}^{n-i}\sin(\varphi _{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=\rho \cos(\varphi _{n-i+1})\prod _{k=1}^{n-i}\sin(\varphi _{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b17fd1be379aa2895fe0f9782f45c0c6094eda57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.44ex; height:7.176ex;" alt="{\displaystyle x_{i}=\rho \cos(\varphi _{n-i+1})\prod _{k=1}^{n-i}\sin(\varphi _{k})}" />.</dd> <dd>On montre par ailleurs que le <a href="/wiki/Jacobien" class="mw-redirect" title="Jacobien">jacobien</a> de la transformation des coordonnées cartésiennes en coordonnées hypersphériques est : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\rho ^{n-1}\prod _{i=1}^{n-2}\sin ^{n-1-i}(\varphi _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\rho ^{n-1}\prod _{i=1}^{n-2}\sin ^{n-1-i}(\varphi _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd6cc6d46de42011cf90bbdc4f1a6a466875367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.101ex; height:7.343ex;" alt="{\displaystyle J=\rho ^{n-1}\prod _{i=1}^{n-2}\sin ^{n-1-i}(\varphi _{i})}" /></dd> <dd>avec φ1 ≤ j ≤ n – 2 ∈ [0,π] et φn – 1 ∈ [0,2π].</dd> <dd>Il en résulte : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\rho ^{n-1}\mathrm {d} \rho \mathrm {d} \varphi _{n-1}\prod _{j=1}^{n-2}\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </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>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\rho ^{n-1}\mathrm {d} \rho \mathrm {d} \varphi _{n-1}\prod _{j=1}^{n-2}\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beea17d2363966f01bd67075453c8026b94e7ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.742ex; height:7.676ex;" alt="{\displaystyle {\hat {f}}({\vec {\tau }})=\int _{\mathbb {R} ^{n}}g(\rho )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\rho ^{n-1}\mathrm {d} \rho \mathrm {d} \varphi _{n-1}\prod _{j=1}^{n-2}\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}" /></dd></dl></dd> <dd>Du fait de la symétrie radiale, on ne change rien de l'intégrale si on considère <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9476c9325cf30de081340cd070b30c3bd93f311a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.333ex; height:2.343ex;" alt="{\displaystyle {\vec {\tau }}}" /> parallèle à l'axe x1. Cela revient alors à avoir θ = φ1 (et indépendant des φj ≠ 1)</dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}\underbrace {\left(\prod _{j=2}^{n-2}\int _{\varphi _{j}=0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}\right)} _{<1>}\left(\int _{0}^{2\pi }\mathrm {d} \varphi _{n-1}\right)}"> <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"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mn>2</mn> <mo>></mo> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mn>1</mn> <mo>></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}\underbrace {\left(\prod _{j=2}^{n-2}\int _{\varphi _{j}=0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}\right)} _{<1>}\left(\int _{0}^{2\pi }\mathrm {d} \varphi _{n-1}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4962435407b57c2a8e0e6b424c847c34675a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:100.308ex; height:11.176ex;" alt="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}\underbrace {\left(\prod _{j=2}^{n-2}\int _{\varphi _{j}=0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}\right)} _{<1>}\left(\int _{0}^{2\pi }\mathrm {d} \varphi _{n-1}\right)}" /> <dl><dd>Calcul de <1></dd> <dd>Posons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13402f22621c46d3eb89c49bee4de32c3aa0ffc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.149ex; height:5.843ex;" alt="{\displaystyle I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}}" /></dd> <dd>On reconnaît ici la <a href="/wiki/Fonction_b%C3%AAta" title="Fonction bêta">fonction bêta</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} (p,q)=2\int _{0}^{\frac {\pi }{2}}\sin ^{2p-1}(\alpha )\cos ^{2q-1}(\alpha )\mathrm {d} \alpha ={\frac {\Gamma (p)\Gamma (q)}{\Gamma (p+q)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</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"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (p,q)=2\int _{0}^{\frac {\pi }{2}}\sin ^{2p-1}(\alpha )\cos ^{2q-1}(\alpha )\mathrm {d} \alpha ={\frac {\Gamma (p)\Gamma (q)}{\Gamma (p+q)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7dbdd2eaed7fc12a4b96fbce789e4cba5e11b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.378ex; height:6.843ex;" alt="{\displaystyle \mathrm {B} (p,q)=2\int _{0}^{\frac {\pi }{2}}\sin ^{2p-1}(\alpha )\cos ^{2q-1}(\alpha )\mathrm {d} \alpha ={\frac {\Gamma (p)\Gamma (q)}{\Gamma (p+q)}}}" /> avec Γ la <a href="/wiki/Fonction_gamma" title="Fonction gamma">fonction gamma</a> et p, q réels positifs.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}={\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}\Rightarrow <1>=\prod _{j=2}^{n-2}{\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}={\frac {\pi ^{\frac {n-3}{2}}}{\Gamma ({\frac {n-1}{2}})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>j</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>j</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒<</mo> <mn>1</mn> <mo>>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>j</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}={\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}\Rightarrow <1>=\prod _{j=2}^{n-2}{\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}={\frac {\pi ^{\frac {n-3}{2}}}{\Gamma ({\frac {n-1}{2}})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ef88c7330af7af7b0be92ea69b258ebfba5dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:82.894ex; height:8.509ex;" alt="{\displaystyle \Rightarrow I_{j}=\int _{0}^{\pi }\sin ^{n-1-j}(\varphi _{j})\mathrm {d} \varphi _{j}={\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}\Rightarrow <1>=\prod _{j=2}^{n-2}{\frac {\Gamma ({\frac {n-j}{2}})\Gamma ({\frac {1}{2}})}{\Gamma ({\frac {n-j+1}{2}})}}={\frac {\pi ^{\frac {n-3}{2}}}{\Gamma ({\frac {n-1}{2}})}}}" /> avec Γ(1) = 1; Γ(1/2) = √π</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n-1}{2}}}{\Gamma ({\frac {n-1}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}}"> <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"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> <mn>2</mn> <mo>></mo> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n-1}{2}}}{\Gamma ({\frac {n-1}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d59a58574ca33f223da9144c25988b867de9a72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; margin-right: -0.028ex; width:66.071ex; height:10.843ex;" alt="{\displaystyle \Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n-1}{2}}}{\Gamma ({\frac {n-1}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho \underbrace {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)} _{<2>}}" /></dd> <dd>On notera au passage <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{0}^{\frac {\pi }{2}}\sin ^{n-2}\alpha \mathrm {d} \alpha ={\frac {\Gamma ({\frac {n-1}{2}}){\sqrt {\pi }}}{\Gamma ({\frac {n}{2}})}}\Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\int _{0}^{\frac {\pi }{2}}\sin ^{n-2}\alpha \mathrm {d} \alpha ={\frac {\Gamma ({\frac {n-1}{2}}){\sqrt {\pi }}}{\Gamma ({\frac {n}{2}})}}\Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e124b2a3ce63ed4433573e4f7750f470897a3951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.633ex; height:7.843ex;" alt="{\displaystyle 2\int _{0}^{\frac {\pi }{2}}\sin ^{n-2}\alpha \mathrm {d} \alpha ={\frac {\Gamma ({\frac {n-1}{2}}){\sqrt {\pi }}}{\Gamma ({\frac {n}{2}})}}\Rightarrow {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}}" /></dd></dl></dd> <dd>Calcul de <2></dd> <dd>Considérons la fonction <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}(z)={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}(z)={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2352626d4abf42701b3df0cdddcab1fd474b9171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.755ex; height:7.509ex;" alt="{\displaystyle L_{n}(z)={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}" /></dd></dl></dd> <dd>On notera Ln(0) = 1</dd> <dd>On a alors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {\int _{0}^{\pi }i\cos \theta \sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>i</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {\int _{0}^{\pi }i\cos \theta \sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb87d54c46b2c9cb08f152e17bfff3ea6fb8be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.275ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {\int _{0}^{\pi }i\cos \theta \sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}" /></dd></dl></dd></dl> En intégrant par parties l'intégrale en numérateur, on établit la relation : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {z}{n-1}}{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {z}{n-1}}{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41fe86d921927fbcdb9169b7c9052ae77e27ed96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.051ex; height:7.509ex;" alt="{\displaystyle -{\frac {\mathrm {d} L_{n}(z)}{\mathrm {d} z}}={\frac {z}{n-1}}{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}" /></dd></dl></dd> <dd>On notera alors : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} L_{n}(0)}{\mathrm {d} z}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} L_{n}(0)}{\mathrm {d} z}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48b30560b7f16bd78d1b85cdb7085cd73e8cac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.163ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} L_{n}(0)}{\mathrm {d} z}}=0}" />.</dd> <dd>En dérivant une seconde fois : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}L_{n}(z)}{dz^{2}}}=-{\frac {\int _{0}^{\pi }\sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}+{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}\Rightarrow {\frac {d^{2}L_{n}(z)}{dz^{2}}}+{\frac {n-1}{z}}{\frac {dL_{n}(z)}{dz}}+L_{n}(z)=0}"> <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"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}L_{n}(z)}{dz^{2}}}=-{\frac {\int _{0}^{\pi }\sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}+{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}\Rightarrow {\frac {d^{2}L_{n}(z)}{dz^{2}}}+{\frac {n-1}{z}}{\frac {dL_{n}(z)}{dz}}+L_{n}(z)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be7a85fcc88ba8c0e6980a30205eaf5464559e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:96.867ex; height:7.509ex;" alt="{\displaystyle {\frac {d^{2}L_{n}(z)}{dz^{2}}}=-{\frac {\int _{0}^{\pi }\sin ^{n-2}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}+{\frac {\int _{0}^{\pi }\sin ^{n}\theta \mathrm {e} ^{\mathrm {i} z\cos \theta }\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}\Rightarrow {\frac {d^{2}L_{n}(z)}{dz^{2}}}+{\frac {n-1}{z}}{\frac {dL_{n}(z)}{dz}}+L_{n}(z)=0}" />.</dd></dl></dd> <dd>On reconnaît ici une équation qui est proche de l'<a href="/wiki/%C3%89quation_diff%C3%A9rentielle_de_Bessel" class="mw-redirect" title="Équation différentielle de Bessel">équation différentielle de Bessel</a>. Pour faire disparaître le facteur n – 1 du deuxième terme, posons : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}(z)=a_{n}z^{-m}J_{m}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}(z)=a_{n}z^{-m}J_{m}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9e4486d3c1f0db3dc0bbf016fcf34e2cf2ac9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.152ex; height:3.009ex;" alt="{\displaystyle L_{n}(z)=a_{n}z^{-m}J_{m}(z)}" />.</dd></dl></dd> <dd>En reportant cette expression dans l'équation différentielle, on arrive à : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}J_{m}(z)}{\mathrm {d} z^{2}}}+\left({\frac {n-1-2m}{z}}\right){\frac {\mathrm {d} J_{m}(z)}{\mathrm {d} z}}+\left({\frac {m(m-n+2)}{z^{2}}}+1\right)J_{m}(z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>m</mi> </mrow> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}J_{m}(z)}{\mathrm {d} z^{2}}}+\left({\frac {n-1-2m}{z}}\right){\frac {\mathrm {d} J_{m}(z)}{\mathrm {d} z}}+\left({\frac {m(m-n+2)}{z^{2}}}+1\right)J_{m}(z)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e8d104c7d1d21a4ff0dfde5bfe913d6a2f2763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.702ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}J_{m}(z)}{\mathrm {d} z^{2}}}+\left({\frac {n-1-2m}{z}}\right){\frac {\mathrm {d} J_{m}(z)}{\mathrm {d} z}}+\left({\frac {m(m-n+2)}{z^{2}}}+1\right)J_{m}(z)=0}" />.</dd></dl></dd> <dd>Il suffit alors de poser m = n – 2/2 pour arriver à l'équation différentielle de Bessel suivante : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}J_{\frac {n-2}{2}}(z)+{\frac {1}{z}}{\frac {\mathrm {d} }{\mathrm {d} z}}J_{\frac {n-2}{2}}(z)+\left(1-\left({\frac {n-2}{2z}}\right)^{2}\right)J_{\frac {n-2}{2}}(z)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}J_{\frac {n-2}{2}}(z)+{\frac {1}{z}}{\frac {\mathrm {d} }{\mathrm {d} z}}J_{\frac {n-2}{2}}(z)+\left(1-\left({\frac {n-2}{2z}}\right)^{2}\right)J_{\frac {n-2}{2}}(z)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79093c0e5ae8baa4a53f6f998757852632d8ef98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.497ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}J_{\frac {n-2}{2}}(z)+{\frac {1}{z}}{\frac {\mathrm {d} }{\mathrm {d} z}}J_{\frac {n-2}{2}}(z)+\left(1-\left({\frac {n-2}{2z}}\right)^{2}\right)J_{\frac {n-2}{2}}(z)=0}" /></dd></dl></dd> <dd>Il s'agit bien d'une équation différentielle de Bessel dont la <a href="/wiki/Fonction_de_Bessel" title="Fonction de Bessel">fonction de Bessel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\frac {n-2}{2}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\frac {n-2}{2}}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8cd7918cd39abd371976a413c469b21d9baad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.762ex; height:4.176ex;" alt="{\displaystyle J_{\frac {n-2}{2}}(z)}" /> est solution. Il en résulte alors la relation suivante, par définition de la fonction de Bessel :</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\frac {n-2}{2}}(z)=\left({\frac {z}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {z^{\frac {n-2}{2}}}{a_{n}}}L_{n}(z)\Rightarrow L_{n}(z)=a_{n}\left({\frac {1}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <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>p</mi> </mrow> </msup> </mrow> <mrow> <mi>p</mi> <mo>!</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <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>p</mi> </mrow> </msup> </mrow> <mrow> <mi>p</mi> <mo>!</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\frac {n-2}{2}}(z)=\left({\frac {z}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {z^{\frac {n-2}{2}}}{a_{n}}}L_{n}(z)\Rightarrow L_{n}(z)=a_{n}\left({\frac {1}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd3b6881e8615b505d644bbad89d60918d5dd7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:102.423ex; height:8.176ex;" alt="{\displaystyle J_{\frac {n-2}{2}}(z)=\left({\frac {z}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {z^{\frac {n-2}{2}}}{a_{n}}}L_{n}(z)\Rightarrow L_{n}(z)=a_{n}\left({\frac {1}{2}}\right)^{\frac {n-2}{2}}\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}}" /></dd> <dd>Avec <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}(0)=1\Rightarrow a_{n}=2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)\Rightarrow L_{n}(z)=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <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>p</mi> </mrow> </msup> </mrow> <mrow> <mi>p</mi> <mo>!</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}(0)=1\Rightarrow a_{n}=2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)\Rightarrow L_{n}(z)=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeebd33b410d35b8862ddc6e6c5de9cfd5bcb7e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:96.622ex; height:7.676ex;" alt="{\displaystyle L_{n}(0)=1\Rightarrow a_{n}=2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)\Rightarrow L_{n}(z)=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\Gamma (p+{\frac {n}{2}})}}\left({\frac {z}{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} z\cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}" /></dd></dl></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow L_{n}(2\pi \rho \tau )=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\,\Gamma (p+{\frac {n}{2}})}}\left({\frac {2\pi \rho \tau }{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} 2\pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <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>p</mi> </mrow> </msup> </mrow> <mrow> <mi>p</mi> <mo>!</mo> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow L_{n}(2\pi \rho \tau )=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\,\Gamma (p+{\frac {n}{2}})}}\left({\frac {2\pi \rho \tau }{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} 2\pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e59eb9cebf63052f86b9ec1284c59222392d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:76.472ex; height:7.676ex;" alt="{\displaystyle \Rightarrow L_{n}(2\pi \rho \tau )=\Gamma \left({\frac {n}{2}}\right)\sum _{p=0}^{+\infty }{\frac {(-1)^{p}}{p!\,\Gamma (p+{\frac {n}{2}})}}\left({\frac {2\pi \rho \tau }{2}}\right)^{2p}={\frac {\int _{0}^{\pi }\mathrm {e} ^{\mathrm {i} 2\pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta }{\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }}}" /></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow <2>=\int _{0}^{\pi }\mathrm {e} ^{2\mathrm {i} \pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta =2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)(2\pi \rho \tau )^{-{\frac {n-2}{2}}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒<</mo> <mn>2</mn> <mo>>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow <2>=\int _{0}^{\pi }\mathrm {e} ^{2\mathrm {i} \pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta =2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)(2\pi \rho \tau )^{-{\frac {n-2}{2}}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10eff2476a36fbed94a6556ed2b73c921332774c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:87.648ex; height:5.843ex;" alt="{\displaystyle \Rightarrow <2>=\int _{0}^{\pi }\mathrm {e} ^{2\mathrm {i} \pi \rho \tau \cos \theta }\sin ^{n-2}\theta \,\mathrm {d} \theta =2^{\frac {n-2}{2}}\Gamma \left({\frac {n}{2}}\right)(2\pi \rho \tau )^{-{\frac {n-2}{2}}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\int _{0}^{\pi }\sin ^{n-2}\theta \,\mathrm {d} \theta }" /></dd></dl> <dl><dd>En revenant à l'expression de la transformée de Fourier : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}=2\pi (\tau )^{\frac {2-n}{2}}\int _{0}^{+\infty }g(\rho )(\rho )^{\frac {n}{2}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\mathrm {d} \rho }"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">i</mi> <mi>π</mi> </mrow> </mrow> <mi>τ</mi> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>ρ</mi> <mi>τ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}=2\pi (\tau )^{\frac {2-n}{2}}\int _{0}^{+\infty }g(\rho )(\rho )^{\frac {n}{2}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\mathrm {d} \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f752f87a79a8fd97a629fc6098336875ddd41a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:101.213ex; height:7.676ex;" alt="{\displaystyle {\hat {f}}({\vec {\tau }})=2{\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}\int _{0}^{+\infty }g(\rho )\rho ^{n-1}\mathrm {d} \rho {\frac {\left(\int _{0}^{\pi }\sin ^{n-2}(\theta )~{\rm {e}}^{{\rm {2i\pi }}\tau \rho \cos \theta }\mathrm {d} \theta \right)}{\int _{0}^{\pi }\sin ^{n-2}\alpha \mathrm {d} \alpha }}=2\pi (\tau )^{\frac {2-n}{2}}\int _{0}^{+\infty }g(\rho )(\rho )^{\frac {n}{2}}J_{\frac {n-2}{2}}(2\pi \rho \tau )\mathrm {d} \rho }" />.</dd></dl></dd></dl> </div><div class="clear" style="clear:both;"></div> </div> Si la transformée de Fourier de f est elle-même une fonction intégrable, on a alors la formule d'inversion : <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\hat {f}}(\xi )~{\rm {e}}^{{\rm {i}}x\cdot \xi }~\mathrm {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> <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> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\hat {f}}(\xi )~{\rm {e}}^{{\rm {i}}x\cdot \xi }~\mathrm {d} \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2184740a035c27fd934a36da5c827ad74614257e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.195ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}{\hat {f}}(\xi )~{\rm {e}}^{{\rm {i}}x\cdot \xi }~\mathrm {d} \xi }" />.</center> Par conséquent, la transformation de Fourier de L1 dans C0 est <a href="/wiki/Injection_(math%C3%A9matiques)" title="Injection (mathématiques)">injective</a> (mais <a href="/wiki/Th%C3%A9or%C3%A8me_de_Banach-Schauder#Exemple_d'application" title="Théorème de Banach-Schauder">pas surjective</a>). <div class="mw-heading mw-heading2"><h2 id="Transformation_de_Fourier_pour_les_fonctions_de_carré_sommable">Transformation de Fourier pour les fonctions de carré sommable</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=8" title="Modifier la section : Transformation de Fourier pour les fonctions de carré sommable" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=8" title="Modifier le code source de la section : Transformation de Fourier pour les fonctions de carré sommable">modifier le code</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Extension_de_la_transformation_de_L1∩L2_à_L2">Extension de la transformation de L1∩L2 à L2</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=9" title="Modifier la section : Extension de la transformation de L1∩L2 à L2" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=9" title="Modifier le code source de la section : Extension de la transformation de L1∩L2 à L2">modifier le code</a>]</div> Le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Plancherel" title="Théorème de Plancherel">théorème de Plancherel</a> permet de donner un sens à la transformée de Fourier des fonctions de <a href="/wiki/Carr%C3%A9_sommable" title="Carré sommable">carré sommable</a> sur ℝ. On commence par un premier résultat préparatoire. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Lemme — Soit h une fonction complexe deux fois continûment dérivable sur ℝ, qui vérifie l'estimation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in \mathbb {R} \quad |h(x)|\leq C/(1+x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in \mathbb {R} \quad |h(x)|\leq C/(1+x^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136fe1e2f3db29456761ca451cf159df92802bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.458ex; height:3.176ex;" alt="{\displaystyle \forall x\in \mathbb {R} \quad |h(x)|\leq C/(1+x^{2})}" /> (où C est une constante),</dd></dl> et dont les deux premières dérivées sont intégrables. Ceci implique que la transformée de Fourier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61505780f3740aa55551090a2b23c668c934a82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.843ex;" alt="{\displaystyle {\hat {h}}}" /> est bien définie et de carré intégrable. De plus, on a l'identité : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>h</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f808b2cd74e86786c8a3bbe2bc4de93450a08ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.084ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }" />.</dd></dl> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Preuve par la <a href="/wiki/Formule_sommatoire_de_Poisson" title="Formule sommatoire de Poisson">formule sommatoire de Poisson</a></div><div class="NavContent" style="padding-bottom:0.4em"> On reprend la formule établie ci-dessus dans la démonstration de la formule d'inversion de Fourier : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in Z}h(x+2\pi n){\rm {e}}^{-2\mathrm {i} \pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k){\rm {e}}^{\mathrm {i} (k+y)x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>Z</mi> </mrow> </munder> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>n</mi> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <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>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in Z}h(x+2\pi n){\rm {e}}^{-2\mathrm {i} \pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k){\rm {e}}^{\mathrm {i} (k+y)x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb83a64ad435e4f6632c6c11ebbbb929ceb543d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.325ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in Z}h(x+2\pi n){\rm {e}}^{-2\mathrm {i} \pi ny}={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }{\hat {h}}(y+k){\rm {e}}^{\mathrm {i} (k+y)x}}" />.</dd></dl></dd></dl> On prend le carré du module des deux membres, et l'on intègre sur l'intervalle [0 ; 1] par rapport à y et sur l'intervalle [0 ; 2π] par rapport à x : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}\int _{0}^{2\pi }\sum _{m,n\in \mathbb {Z} }h(x+2\pi n){\bar {h}}(x+2\pi m){\rm {e}}^{2\mathrm {i} \pi (m-n)y}\,{\rm {d}}x\,{\rm {d}}y={\frac {1}{4\pi ^{2}}}\int _{0}^{1}\int _{0}^{2\pi }\sum _{j,k\in \mathbb {Z} }{\hat {h}}(y+j){\bar {\hat {h}}}(y+k){\rm {e}}^{\mathrm {i} (j-k)x}\,{\rm {d}}x\,{\rm {d}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">)</mo> <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>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>m</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <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>y</mi> <mo>+</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}\int _{0}^{2\pi }\sum _{m,n\in \mathbb {Z} }h(x+2\pi n){\bar {h}}(x+2\pi m){\rm {e}}^{2\mathrm {i} \pi (m-n)y}\,{\rm {d}}x\,{\rm {d}}y={\frac {1}{4\pi ^{2}}}\int _{0}^{1}\int _{0}^{2\pi }\sum _{j,k\in \mathbb {Z} }{\hat {h}}(y+j){\bar {\hat {h}}}(y+k){\rm {e}}^{\mathrm {i} (j-k)x}\,{\rm {d}}x\,{\rm {d}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81804984cdbbc075a76fa657d115a3ddd1486a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:102.666ex; height:7.176ex;" alt="{\displaystyle \int _{0}^{1}\int _{0}^{2\pi }\sum _{m,n\in \mathbb {Z} }h(x+2\pi n){\bar {h}}(x+2\pi m){\rm {e}}^{2\mathrm {i} \pi (m-n)y}\,{\rm {d}}x\,{\rm {d}}y={\frac {1}{4\pi ^{2}}}\int _{0}^{1}\int _{0}^{2\pi }\sum _{j,k\in \mathbb {Z} }{\hat {h}}(y+j){\bar {\hat {h}}}(y+k){\rm {e}}^{\mathrm {i} (j-k)x}\,{\rm {d}}x\,{\rm {d}}y}" />.</dd></dl> On peut échanger l'ordre de la sommation et des deux intégrations dans l'expression ci-dessus, parce que les hypothèses faites sur h impliquent que les séries convergent normalement dans l'espace des fonctions continues de x et y, périodiques de période 2π en x et de période 1 en y. L'intégration en y du premier membre ne laisse subsister que les termes pour lesquels m et n sont égaux, et l'intégration en x du deuxième membre ne laisse subsister que les termes pour lesquels j et k sont identiques. Il reste donc : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in \mathbb {Z} }\int _{0}^{2\pi }|h(x+2\pi n)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }\int _{0}^{1}|{\hat {h}}(y+k)|^{2}\,{\rm {d}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>n</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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>y</mi> <mo>+</mo> <mi>k</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }\int _{0}^{2\pi }|h(x+2\pi n)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }\int _{0}^{1}|{\hat {h}}(y+k)|^{2}\,{\rm {d}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53b4cd12b92a5434e0621de8f8b0b42dff0c45b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.156ex; height:7.009ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }\int _{0}^{2\pi }|h(x+2\pi n)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\sum _{k\in \mathbb {Z} }\int _{0}^{1}|{\hat {h}}(y+k)|^{2}\,{\rm {d}}y}" />.</dd></dl> Il suffit de faire dans le premier membre le changement de variable dans chaque intégrale x + 2π n = x' et dans le second le changement de variable dans chaque intégrale y + k = ξ, et l'on obtient la formule : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }|h(x')|^{2}\,{\rm {d}}x'={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>h</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }|h(x')|^{2}\,{\rm {d}}x'={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37db1cbc3d4f02df5d6a204b548b4e57ec53758e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.454ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }|h(x')|^{2}\,{\rm {d}}x'={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }" />.</dd></dl></dd></dl> Après changement de la variable muette x' en x, on obtient la formule annoncée. </div><div class="clear" style="clear:both;"></div> </div> Une fois démontrée dans le lemme ci-dessus la formule de Plancherel pour une classe de fonctions suffisamment régulières, on étend par densité la transformation de Fourier à tout L2(ℝ). <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Extension de la transformation de Fourier par densité</div><div class="NavContent" style="padding-bottom:0.4em"> On adopte encore les mêmes notations que dans la démonstration de la formule d'inversion de Fourier par la formule sommatoire de Poisson, donc ϕ est une fonction deux fois continûment différentiable, à support compact, et d'intégrale 1. On pose ϕp(x) =p ϕ(px). Soit h une fonction de carré intégrable, et soit p un nombre entier quelconque. On définit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{p}=(h1_{[-p,p]})*\phi _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>h</mi> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>p</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{p}=(h1_{[-p,p]})*\phi _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efca151b901e3173c84a514d26abb06d496d7359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.983ex; height:3.176ex;" alt="{\displaystyle h_{p}=(h1_{[-p,p]})*\phi _{p}}" /></dd></dl> et l'on peut montrer le résultat suivant : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|h-h_{p}|^{2}\,{\rm {d}}x=0}"> <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>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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>h</mi> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|h-h_{p}|^{2}\,{\rm {d}}x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8830e51f3cb559c1edefbda989f4bd1dce77ad41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.063ex; width:23.858ex; height:5.676ex;" alt="{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|h-h_{p}|^{2}\,{\rm {d}}x=0}" />.</dd></dl> La démonstration utilise des techniques classiques d'approximation par régularisation. D'autre part, les fonctions hp ont les propriétés nécessaires pour appliquer le lemme ci-dessus, et en particulier <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }|h_{p}-h_{q}|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}_{p}-{\hat {h}}_{q}|^{2}\,{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }|h_{p}-h_{q}|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}_{p}-{\hat {h}}_{q}|^{2}\,{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b088435d57f321aca7a30cd43bab30d6b1797e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.56ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }|h_{p}-h_{q}|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}_{p}-{\hat {h}}_{q}|^{2}\,{\rm {d}}\xi }" />.</dd></dl></dd></dl> Comme la suite (hp)p ≥ 1 est de Cauchy dans l'espace L2(ℝ), la suite des transformées de Fourier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {h}}_{p})_{p\geq 1}}"> <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>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {h}}_{p})_{p\geq 1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e49b37de9c3a404cb2abc9bba1f9a5212d10349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.367ex; height:3.509ex;" alt="{\displaystyle ({\hat {h}}_{p})_{p\geq 1}}" /> est aussi de Cauchy, donc elle converge. Sa limite, que l'on note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61505780f3740aa55551090a2b23c668c934a82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.843ex;" alt="{\displaystyle {\hat {h}}}" />, ne dépend pas du choix de la suite d'approximations. En effet, si gp était une autre suite d'approximations convergeant vers h en moyenne quadratique, et satisfaisant les conditions fonctionnelles sous lesquelles on peut appliquer la formule sommatoire de Poisson, on aurait l'estimation <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|g_{p}-h_{p}\|_{2}\leq \|g_{p}-h\|_{2}+\|h-h_{p}\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <mi>h</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>h</mi> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|g_{p}-h_{p}\|_{2}\leq \|g_{p}-h\|_{2}+\|h-h_{p}\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90a4eb254ea1bb7fe2b0c4bdc89cabb92b1285c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.408ex; height:3.009ex;" alt="{\displaystyle \|g_{p}-h_{p}\|_{2}\leq \|g_{p}-h\|_{2}+\|h-h_{p}\|_{2}}" />,</dd></dl></dd></dl> qui tend vers 0 pour p tendant vers l'infini. Par conséquent, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\hat {g}}_{p}-{\hat {h}}_{p}\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</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>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\hat {g}}_{p}-{\hat {h}}_{p}\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1764de5e7e2c2e38e72d65880d6dc065d2d2cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.908ex; height:3.676ex;" alt="{\displaystyle \|{\hat {g}}_{p}-{\hat {h}}_{p}\|_{2}}" /> tend aussi vers 0 et l'on conclut que la limite de la suite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191b00611e357aff404af748727f6893a00e3a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.291ex; height:3.009ex;" alt="{\displaystyle {\hat {g}}_{p}}" /> est bien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {h}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61505780f3740aa55551090a2b23c668c934a82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.843ex;" alt="{\displaystyle {\hat {h}}}" />. </div><div class="clear" style="clear:both;"></div> </div> On a ainsi le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Plancherel" title="Théorème de Plancherel">théorème de Plancherel</a> : <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Théorème de Plancherel —  Soit f une fonction complexe sur ℝ et de carré sommable. Alors la transformée de Fourier de f peut être définie comme suit : pour tout p entier, on pose <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{p}(x)=(f1_{[-p,p]})(x)={\begin{cases}f(x)&{\text{si }}|x|\leq p,\\0&{\text{sinon.}}\end{cases}}}"> <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> <mo stretchy="false">(</mo> <mi>f</mi> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>p</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>si </mtext> </mrow> <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>p</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>sinon.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{p}(x)=(f1_{[-p,p]})(x)={\begin{cases}f(x)&{\text{si }}|x|\leq p,\\0&{\text{sinon.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6962d1ac39bc831aba24e5b98747d91d557d29f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.377ex; height:6.176ex;" alt="{\displaystyle f_{p}(x)=(f1_{[-p,p]})(x)={\begin{cases}f(x)&{\text{si }}|x|\leq p,\\0&{\text{sinon.}}\end{cases}}}" /></dd></dl> La suite des transformées de Fourier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{p}}"> <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>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd02efc4a0a086e89f480bc74d7ad9a2c9f0452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.758ex; height:3.676ex;" alt="{\displaystyle {\hat {f}}_{p}}" /> converge dans L2(ℝ), et sa limite est la transformée de Fourier <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}}}" />, c'est-à-dire <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|{\hat {f}}(\xi )-{\hat {f}}_{p}(\xi )|^{2}\,{\rm {d}}\xi =0}"> <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>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <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"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ξ</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|{\hat {f}}(\xi )-{\hat {f}}_{p}(\xi )|^{2}\,{\rm {d}}\xi =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d585ba3213a7928b51555983c233ee69ccb5993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.063ex; width:29.957ex; height:5.676ex;" alt="{\displaystyle \lim _{p\to \infty }\int _{\mathbb {R} }|{\hat {f}}(\xi )-{\hat {f}}_{p}(\xi )|^{2}\,{\rm {d}}\xi =0}" />.</dd></dl> De plus, on a l'identité : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }|f(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {f}}(\xi )|^{2}{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }|f(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {f}}(\xi )|^{2}{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/157b2188f88bd9763425adecc35190b59d835748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.997ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }|f(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {f}}(\xi )|^{2}{\rm {d}}\xi }" />.</dd></dl> De façon similaire, si l'on pose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{p}(x)=\int _{-p}^{p}{\hat {f}}(\xi ){\rm {e}}^{\mathrm {i} x\xi }\,{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{p}(x)=\int _{-p}^{p}{\hat {f}}(\xi ){\rm {e}}^{\mathrm {i} x\xi }\,{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92feb24708faae96dc00d1b41b93a0264640621a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.062ex; height:6.176ex;" alt="{\displaystyle g_{p}(x)=\int _{-p}^{p}{\hat {f}}(\xi ){\rm {e}}^{\mathrm {i} x\xi }\,{\rm {d}}\xi }" />, les gp convergent en moyenne quadratique vers f. </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration du théorème de Plancherel</div><div class="NavContent" style="padding-bottom:0.4em"> L'identité suivante résulte du procédé d'extension décrit ci-dessus : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>h</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f808b2cd74e86786c8a3bbe2bc4de93450a08ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.084ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} }|h(x)|^{2}\,{\rm {d}}x={\frac {1}{2\pi }}\int _{\mathbb {R} }|{\hat {h}}(\xi )|^{2}\,{\rm {d}}\xi }" />.</dd></dl> Considérons alors la suite de fonctions fp = f 1{[–p,p]} . En vertu du théorème de convergence dominée de Lebesgue pour les fonctions de <a href="/wiki/Carr%C3%A9_sommable" title="Carré sommable">carré sommable</a>, la suite des fp converge en moyenne quadratique vers f et par conséquent, on aura aussi <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{p\to \infty }\|{\hat {f}}-{\hat {f}}_{p}\|_{2}=0}"> <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>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo fence="false" 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>−</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>p</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to \infty }\|{\hat {f}}-{\hat {f}}_{p}\|_{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb0955357bd96ce2a5aa66e122576b4f952a167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.063ex; width:19.501ex; height:4.676ex;" alt="{\displaystyle \lim _{p\to \infty }\|{\hat {f}}-{\hat {f}}_{p}\|_{2}=0}" />.</dd></dl> En d'autres termes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}_{p}}"> <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>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd02efc4a0a086e89f480bc74d7ad9a2c9f0452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.758ex; height:3.676ex;" alt="{\displaystyle {\hat {f}}_{p}}" /> converge en moyenne quadratique vers <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}}}" />. La démonstration pour la formule d'inversion est analogue. </div><div class="clear" style="clear:both;"></div> </div> Ainsi la transformation de Fourier-Plancherel définit un <a href="/wiki/Automorphisme" title="Automorphisme">automorphisme</a> intemporel de l'espace L2, qui est une <a href="/wiki/Isom%C3%A9trie" title="Isométrie">isométrie</a>, à condition de faire un changement d'échelle si l'on utilise la notation en pulsation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\hat {f}}/{\sqrt {2\pi }}\|_{2}=\|f{\sqrt {2\pi }}\|_{2}}"> <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>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\hat {f}}/{\sqrt {2\pi }}\|_{2}=\|f{\sqrt {2\pi }}\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba43a47f2198970dc5fcb7cc99ad81dd552953e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.858ex; height:3.343ex;" alt="{\displaystyle \|{\hat {f}}/{\sqrt {2\pi }}\|_{2}=\|f{\sqrt {2\pi }}\|_{2}}" />.</dd></dl> En physique, on interprète le terme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\hat {f}}(\xi )/{\sqrt {2\pi }}|^{2}}"> <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"> <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"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\hat {f}}(\xi )/{\sqrt {2\pi }}|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0630dfb3114692dfaaced92a37c68e6c2f222e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.479ex; height:3.343ex;" alt="{\displaystyle |{\hat {f}}(\xi )/{\sqrt {2\pi }}|^{2}}" /> figurant sous l'intégrale comme une <a href="/wiki/Densit%C3%A9_spectrale_de_puissance" title="Densité spectrale de puissance">densité spectrale de puissance</a>. La définition de la transformation de Fourier-Plancherel est compatible avec la définition habituelle de la transformée de Fourier des fonctions <a href="/wiki/Int%C3%A9grabilit%C3%A9" title="Intégrabilité">intégrables</a>. Sur l'intersection L1(ℝ)} ∩ L2(ℝ) des domaines de définition, on montre à l'aide du <a href="/wiki/Th%C3%A9or%C3%A8me_de_convergence_domin%C3%A9e" title="Théorème de convergence dominée">théorème de convergence dominée</a> de Lebesgue que les deux définitions coïncident. <div class="mw-heading mw-heading3"><h3 id="La_transformation_vue_comme_opérateur_de_L2(ℝ)">La transformation vue comme opérateur de L2(ℝ)</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=10" title="Modifier la section : La transformation vue comme opérateur de L2(ℝ)" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=10" title="Modifier le code source de la section : La transformation vue comme opérateur de L2(ℝ)">modifier le code</a>]</div> Remarque : ce paragraphe utilise la <a href="#Conventions_alternatives">définition fréquentielle de la transformée de Fourier</a>, pour des raisons d'isométrie. Nous venons de voir que la transformation de Fourier <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}}}" /> induit sur l'<a href="/wiki/Espace_de_Hilbert" title="Espace de Hilbert">espace de Hilbert</a> L2(ℝ) un opérateur linéaire. Nous en récapitulons ici les propriétés : <ul><li><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}}}" /> est un <a href="/wiki/Op%C3%A9rateur_unitaire" title="Opérateur unitaire">opérateur unitaire</a> de L2. Il s'agit en particulier d'une isométrie. On retrouve le premier fait, connu sous le nom de formule de Parseval, affirmant que pour toutes fonctions f, g ∈ L2(ℝ),</li></ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f\mid g\rangle =\langle {\mathcal {F}}f\mid {\mathcal {F}}g\rangle ,\quad {\text{i.e. }}\quad \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,{\rm {d}}x=\int _{-\infty }^{\infty }{\hat {f}}(\nu ){\overline {{\hat {g}}(\nu )}}\,{\rm {d}}\nu }"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" 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>f</mi> <mo>∣</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 fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>i.e. </mtext> </mrow> <mspace width="1em"></mspace> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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>ν</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>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f\mid g\rangle =\langle {\mathcal {F}}f\mid {\mathcal {F}}g\rangle ,\quad {\text{i.e. }}\quad \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,{\rm {d}}x=\int _{-\infty }^{\infty }{\hat {f}}(\nu ){\overline {{\hat {g}}(\nu )}}\,{\rm {d}}\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5dc8aea89506fea04d2c69c806d2292be25b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.456ex; height:6.009ex;" alt="{\displaystyle \langle f\mid g\rangle =\langle {\mathcal {F}}f\mid {\mathcal {F}}g\rangle ,\quad {\text{i.e. }}\quad \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,{\rm {d}}x=\int _{-\infty }^{\infty }{\hat {f}}(\nu ){\overline {{\hat {g}}(\nu )}}\,{\rm {d}}\nu }" /> et en particulier le deuxième fait, connu sous le nom de <a href="/wiki/Th%C3%A9or%C3%A8me_de_Plancherel" title="Théorème de Plancherel">théorème de Plancherel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{2}=\|{\hat {f}}\|_{2},\quad {\text{i.e. }}\quad \int _{-\infty }^{+\infty }|f(x)|^{2}\,{\rm {d}}x=\int _{-\infty }^{+\infty }|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }"> <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>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>i.e. </mtext> </mrow> <mspace width="1em"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</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>ν</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{2}=\|{\hat {f}}\|_{2},\quad {\text{i.e. }}\quad \int _{-\infty }^{+\infty }|f(x)|^{2}\,{\rm {d}}x=\int _{-\infty }^{+\infty }|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269811e96faa1e33e7ee6f9dadd511f914cfe286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.338ex; height:6.176ex;" alt="{\displaystyle \|f\|_{2}=\|{\hat {f}}\|_{2},\quad {\text{i.e. }}\quad \int _{-\infty }^{+\infty }|f(x)|^{2}\,{\rm {d}}x=\int _{-\infty }^{+\infty }|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }" /> ; <ul><li>son inverse (qui est aussi son <a href="/wiki/Op%C3%A9rateur_adjoint" title="Opérateur adjoint">adjoint</a>) est donné par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{-1}g={\mathcal {F}}{\check {g}}}"> <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"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>g</mi> <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-ORD"> <mover> <mi>g</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{-1}g={\mathcal {F}}{\check {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d76be62b07a44cb540f8aa707301fbe5aaa0e8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.709ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}^{-1}g={\mathcal {F}}{\check {g}}}" /> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {g}}:x\mapsto g(-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {g}}:x\mapsto g(-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4527c2c85123271030773198caf904f467e7efcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.176ex; height:2.843ex;" alt="{\displaystyle {\check {g}}:x\mapsto g(-x)}" /> ;</li> <li>en tant qu'<a href="/wiki/Automorphisme" title="Automorphisme">automorphisme</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}}}" /> est de <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">période</a> 4. Autrement dit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{4}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29b8e318d48132aebf6a36b82cfdafe8165d5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.058ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}^{4}}" /> = <a href="/wiki/Application_identit%C3%A9" title="Application identité">id</a> ;</li> <li>en tant qu'<a href="/wiki/Endomorphisme" title="Endomorphisme">endomorphisme</a> de L2(ℝ), <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 pour <a href="/wiki/Valeurs_propres" class="mw-redirect" title="Valeurs propres">valeurs propres</a> les quatre <a href="/wiki/Racine_de_l%27unit%C3%A9" title="Racine de l'unité">racines quatrièmes de l'unité</a> : 1, i, –1 et –i. Une <a href="/wiki/Base_hilbertienne" class="mw-redirect" title="Base hilbertienne">base hilbertienne</a> de vecteurs propres est donnée par les <a href="/wiki/Polyn%C3%B4me_d%27Hermite#Fonctions_d'Hermite-Gauss" title="Polynôme d'Hermite">fonctions d'Hermite-Gauss</a></li></ul> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phi }_{n}(x)={\frac {2^{1/4}}{\sqrt {n!}}}\,{\rm {e}}^{-\pi x^{2}}H_{n}(2x{\sqrt {\pi }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <msqrt> <mi>n</mi> <mo>!</mo> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phi }_{n}(x)={\frac {2^{1/4}}{\sqrt {n!}}}\,{\rm {e}}^{-\pi x^{2}}H_{n}(2x{\sqrt {\pi }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f551c50f294e177dafe63b40e6c4506b69da21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.018ex; height:6.843ex;" alt="{\displaystyle {\phi }_{n}(x)={\frac {2^{1/4}}{\sqrt {n!}}}\,{\rm {e}}^{-\pi x^{2}}H_{n}(2x{\sqrt {\pi }})}" />, où Hn(x) sont les <a href="/wiki/Polyn%C3%B4mes_d%27Hermite" class="mw-redirect" title="Polynômes d'Hermite">polynômes d'Hermite</a> « probabilistes », qui s'écrivent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(x)=(-1)^{n}{\rm {e}}^{\frac {x^{2}}{2}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}{\rm {e}}^{-{\frac {x^{2}}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=(-1)^{n}{\rm {e}}^{\frac {x^{2}}{2}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}{\rm {e}}^{-{\frac {x^{2}}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455fbc303f5ae454ff3e40b952f7e1beeb57ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.67ex; height:5.676ex;" alt="{\displaystyle H_{n}(x)=(-1)^{n}{\rm {e}}^{\frac {x^{2}}{2}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}{\rm {e}}^{-{\frac {x^{2}}{2}}}}" />. Avec ces notations, la formule suivante récapitule la situation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\phi }}_{n}(\nu )=(-{\rm {i}})^{n}{\phi }_{n}(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\phi }}_{n}(\nu )=(-{\rm {i}})^{n}{\phi }_{n}(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4cf57180393c80ed4157dabba775b73a470c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.954ex; height:3.343ex;" alt="{\displaystyle {\hat {\phi }}_{n}(\nu )=(-{\rm {i}})^{n}{\phi }_{n}(\nu )}" />. On retrouve la <a href="/wiki/Fonction_de_Gauss" class="mw-redirect" title="Fonction de Gauss">gaussienne</a> comme première fonction d'Hermite. Ces fonctions appartiennent à la <a href="#Transformation_de_Fourier_sur_l'espace_de_Schwartz">classe de Schwartz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" />. <div class="mw-heading mw-heading2"><h2 id="Lien_avec_le_produit_de_convolution">Lien avec le produit de convolution</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=11" title="Modifier la section : Lien avec le produit de convolution" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=11" title="Modifier le code source de la section : Lien avec le produit de convolution">modifier le code</a>]</div> La transformation de Fourier a des propriétés très intéressantes liées au <a href="/wiki/Produit_de_convolution" title="Produit de convolution">produit de convolution</a>. On rappelle que (d'après l'<a href="/wiki/In%C3%A9galit%C3%A9_de_Young_pour_la_convolution" title="Inégalité de Young pour la convolution">inégalité de Young pour la convolution</a>) : <ul><li>si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5755be3dbed37853fac5883888f75bbc33d896b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.955ex; height:3.176ex;" alt="{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}" />, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a11002d5ae360e3d9ebdaf36fb26069c8b6b0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.116ex; height:3.176ex;" alt="{\displaystyle f*g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f*g\|_{1}\leq \|f\|_{1}\cdot \|g\|_{1}}"> <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"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <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> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f*g\|_{1}\leq \|f\|_{1}\cdot \|g\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea129ac2e348390af15cdcfdb57756418f75bac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.899ex; height:2.843ex;" alt="{\displaystyle \|f*g\|_{1}\leq \|f\|_{1}\cdot \|g\|_{1}}" /> ;</li> <li>si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefd34ceaef10d24ce0ce6dfdcdf62ace13352c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.805ex; height:3.176ex;" alt="{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{N})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a017274075e32f4f98a7f01f5ab3a0f52582ae58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.643ex; height:3.176ex;" alt="{\displaystyle g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}" />, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6022ece9eb6dd5c16ba4c1061ecbf3a183ad2696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.116ex; height:3.176ex;" alt="{\displaystyle f*g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f*g\|_{2}\leq \|f\|_{1}\cdot \|g\|_{2}}"> <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"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <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> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f*g\|_{2}\leq \|f\|_{1}\cdot \|g\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c61e9e03b8ef3ea25258e9739915d835cb074c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.899ex; height:2.843ex;" alt="{\displaystyle \|f*g\|_{2}\leq \|f\|_{1}\cdot \|g\|_{2}}" /> ;</li> <li>si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7013b79861f0cbcee2cf3cb64fe5eb736ea643ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.955ex; height:3.176ex;" alt="{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}" />, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g\in {\rm {L}}^{\infty }(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g\in {\rm {L}}^{\infty }(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363730b625e3eefe0ecc32b1a251d6bea7f610c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.937ex; height:3.176ex;" alt="{\displaystyle f*g\in {\rm {L}}^{\infty }(\mathbb {R} ^{N})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f*g\|_{\infty }\leq \|f\|_{2}\cdot \|g\|_{2}}"> <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"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f*g\|_{\infty }\leq \|f\|_{2}\cdot \|g\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e384cc2236c6b0aa7647d976d4c640440e0626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.72ex; height:2.843ex;" alt="{\displaystyle \|f*g\|_{\infty }\leq \|f\|_{2}\cdot \|g\|_{2}}" />.</li></ul> Ainsi : <ul><li>si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5755be3dbed37853fac5883888f75bbc33d896b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.955ex; height:3.176ex;" alt="{\displaystyle f,g\in {\rm {L}}^{1}(\mathbb {R} ^{N})}" />, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(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 stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</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> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <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>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3489203fdbf4f8eef8e6cb8ebb1a46eb7c904b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.743ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)}" /> ;</li> <li>par densité, cette égalité tient encore si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb4d83b93e0181ded809577d113c74d3f6abba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.626ex; height:3.009ex;" alt="{\displaystyle f\in {\rm {L}}^{1}}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in {\rm {L}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in {\rm {L}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef36ecafbc06a2f614ab840e3080546f2d8eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.464ex; height:3.009ex;" alt="{\displaystyle g\in {\rm {L}}^{2}}" /> ;</li> <li>Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7013b79861f0cbcee2cf3cb64fe5eb736ea643ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.955ex; height:3.176ex;" alt="{\displaystyle f,g\in {\rm {L}}^{2}(\mathbb {R} ^{N})}" />, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\ast g={\mathcal {F}}^{-1}[{\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</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 stretchy="false">[</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> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <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>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\ast g={\mathcal {F}}^{-1}[{\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85b7f0861fdbebc29f4a526712ed8f62f06363e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.637ex; height:3.176ex;" alt="{\displaystyle f\ast g={\mathcal {F}}^{-1}[{\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)]}" /> ; de plus, l'égalité <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(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 stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</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> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <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>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3489203fdbf4f8eef8e6cb8ebb1a46eb7c904b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.743ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f*g)={\mathcal {F}}(f)\,\cdot \,{\mathcal {F}}(g)}" /> est vraie si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g\in {\rm {L}}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g\in {\rm {L}}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea978d10a3dfe60e0b7082ce86be3306c4d5fd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.937ex; height:3.009ex;" alt="{\displaystyle f*g\in {\rm {L}}^{1}}" />.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Principe_d'incertitude">Principe d'incertitude</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=12" title="Modifier la section : Principe d'incertitude" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=12" title="Modifier le code source de la section : Principe d'incertitude">modifier le code</a>]</div> Remarque : ce paragraphe utilise la <a href="#Conventions_alternatives">définition fréquentielle de la transformée de Fourier</a>. <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Principe_d%27incertitude" title="Principe d'incertitude">Principe d'incertitude</a>.</div></div> On peut remarquer que les répartitions d'une fonction et de sa transformée de Fourier ont des comportements opposés : plus la masse de f(x) est « concentrée », plus celle de la transformée est étalée, et inversement. Il est en fait impossible de concentrer à la fois la masse d'une fonction et celle de sa transformée. Ce compromis entre la compaction d'une fonction et celle de sa transformée de Fourier peut se formaliser par un <a href="/wiki/Principe_d%27incertitude" title="Principe d'incertitude">principe d'incertitude</a> en considérant une fonction et sa transformée de Fourier comme des variables conjuguées par la <a href="/wiki/Forme_symplectique" title="Forme symplectique">forme symplectique</a> sur le domaine temps-fréquence : par la <a href="/wiki/Transformation_canonique" title="Transformation canonique">transformation canonique</a> linéaire, la transformation de Fourier est une rotation de 90° dans le domaine temps–fréquence qui préserve la forme symplectique. Supposons f intégrable et de carré intégrable. <a href="/wiki/Sans_perte_de_g%C3%A9n%C3%A9ralit%C3%A9" title="Sans perte de généralité">Sans perte de généralité</a>, on supposera f normalisée : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,{\rm {d}}x=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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,{\rm {d}}x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba2acca0bef9ac20199812370511a09961090a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.869ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,{\rm {d}}x=1}" />.</dd></dl> Par le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Plancherel" title="Théorème de Plancherel">théorème de Plancherel</a>, on sait que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu )}"> <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>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e41aa1f234052893a06bd98fd958d5f349d42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\nu )}" /> est également normalisée. On peut mesurer la répartition autour d'un point (x = 0 sans perte de généralité) par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,{\rm {d}}x}"> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,{\rm {d}}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7389918d63a56db8fce53c8a14cfe07070597b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.157ex; height:6.009ex;" alt="{\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,{\rm {d}}x}" />.</dd></dl> De même pour la fréquence autour du point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7229c47b5bc20ef0a1371a4f3c09459ccb6909ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =0}" /> : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}({\hat {f}})=\int _{-\infty }^{\infty }\nu ^{2}|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }"> <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> <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> <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>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</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>ν</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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}({\hat {f}})=\int _{-\infty }^{\infty }\nu ^{2}|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a646bb000cdbd6917098d73dac571addda50c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.731ex; height:6.009ex;" alt="{\displaystyle D_{0}({\hat {f}})=\int _{-\infty }^{\infty }\nu ^{2}|{\hat {f}}(\nu )|^{2}\,{\rm {d}}\nu }" />.</dd></dl> En probabilités, il s'agit des <a href="/wiki/Moment_(math%C3%A9matiques)" class="mw-redirect" title="Moment (mathématiques)">moments d'ordre 2</a> de |f|2 et de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\hat {f}}|^{2}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\hat {f}}|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5710846c6f8bfac53fa6375ae0d8e1815c445cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.047ex; height:3.343ex;" alt="{\displaystyle |{\hat {f}}|^{2}}" />. Le principe d'incertitude dit que si f(x) est absolument continue et que les fonctions x·f(x) et f′(x) sont de carrés intégrables, on a alors<a href="#cite_note-4">[4]</a> : <dl><dd><math 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> </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/504110dc0487fd928af199bd92324eb25b669af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.201ex; height:5.509ex;" alt="{\displaystyle D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}}" />.</dd></dl> Cette inégalité est aussi connue sous le nom d'inégalité de Heisenberg-Gabor ou simplement <a href="/wiki/Principe_d%27incertitude" title="Principe d'incertitude">inégalité de Heisenberg</a> par son utilisation répandue en <a href="/wiki/M%C3%A9canique_quantique" title="Mécanique quantique">mécanique quantique</a>. L'égalité n'est atteinte que pour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=C_{1}\,{\rm {e}}^{{-\pi x^{2}}/{\sigma ^{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> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=C_{1}\,{\rm {e}}^{{-\pi x^{2}}/{\sigma ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223c64f3ab46fabadf85b5116a17cb9db380fcb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.47ex; height:3.509ex;" alt="{\displaystyle f(x)=C_{1}\,{\rm {e}}^{{-\pi x^{2}}/{\sigma ^{2}}}}" /> (alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=\sigma C_{1}\,{\rm {e}}^{-\pi \sigma ^{2}\xi ^{2}}}"> <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>σ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=\sigma C_{1}\,{\rm {e}}^{-\pi \sigma ^{2}\xi ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/947ba6814f9b73088214f36825133025cd26c919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.89ex; height:3.509ex;" alt="{\displaystyle {\hat {f}}(\xi )=\sigma C_{1}\,{\rm {e}}^{-\pi \sigma ^{2}\xi ^{2}}}" />) pour σ > 0 arbitraire et C1 telle que f est L2–normalisée, soit, si f est une <a href="/wiki/Fonction_gaussienne" title="Fonction gaussienne">fonction gaussienne</a> (normalisée) centrée en 0 et de variance σ2, et sa transformée de Fourier est une gaussienne de variance σ–2. <div class="mw-heading mw-heading2"><h2 id="Transformation_de_Fourier_sur_l'espace_de_Schwartz">Transformation de Fourier sur l'espace de Schwartz</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=13" title="Modifier la section : Transformation de Fourier sur l'espace de Schwartz" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=13" title="Modifier le code source de la section : Transformation de Fourier sur l'espace de Schwartz">modifier le code</a>]</div> L'<a href="/wiki/Espace_de_Schwartz" title="Espace de Schwartz">espace de Schwartz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0078d18e4675b6e7e2acb6c2c25c65294193e36d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" /> est l'espace des fonctions f de classe C∞ sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <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> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" />, telles que f et toutes ses dérivées soient à décroissance rapide. C'est un <a href="/wiki/Sous-espace_vectoriel" title="Sous-espace vectoriel">sous-espace vectoriel</a> de L1, donc pour lequel la transformée de Fourier est définie. Ces fonctions sont à la fois temporellement et fréquentiellement à <a href="/wiki/D%C3%A9croissance_exponentielle" title="Décroissance exponentielle">décroissance exponentielle</a>. L'intérêt de la classe de Schwartz résulte de la propriété d'échange entre régularité et décroissance à l'infini qu'opère la transformée de Fourier. <ul><li>Toute fonction de Schwartz est de classe C∞ avec des dérivées toutes intégrables. On en déduit que sa transformée de Fourier est à décroissance rapide.</li> <li>Toute fonction de Schwartz est à décroissance rapide. On en déduit que sa transformée de Fourier est de classe C∞.</li></ul> Ainsi, on visualise intuitivement pourquoi l'espace de Schwartz est invariant par transformation de Fourier. Cet espace est donc très commode pour l'utilisation de cette dernière. De plus, l'espace de Schwartz est <a href="/wiki/Partie_dense" title="Partie dense">dense</a> dans L1 et dans L2, et pourrait donc servir de base pour la définition de la transformation de Fourier sur ces espaces. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Formule d'inversion de Fourier sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0078d18e4675b6e7e2acb6c2c25c65294193e36d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" /> —  La transformée de Fourier induit un <a href="/wiki/Automorphisme" title="Automorphisme">automorphisme</a> <a href="/wiki/Hom%C3%A9omorphisme" title="Homéomorphisme">bicontinu</a> de l'espace de Schwartz sur lui-même, dont l'inverse est défini par<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {F}}^{-1}\phi )(x)=({\mathcal {F}}\phi )(-x)=\int _{\mathbb {R} ^{n}}\phi (\xi )\,\mathrm {e} ^{2{\rm {i}}\pi x\cdot \xi }\,\mathrm {d} \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</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> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</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>ϕ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>x</mi> <mo>⋅</mo> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}^{-1}\phi )(x)=({\mathcal {F}}\phi )(-x)=\int _{\mathbb {R} ^{n}}\phi (\xi )\,\mathrm {e} ^{2{\rm {i}}\pi x\cdot \xi }\,\mathrm {d} \xi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d840023a9adfbe41879675c4617f1c863de88955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.932ex; height:5.676ex;" alt="{\displaystyle ({\mathcal {F}}^{-1}\phi )(x)=({\mathcal {F}}\phi )(-x)=\int _{\mathbb {R} ^{n}}\phi (\xi )\,\mathrm {e} ^{2{\rm {i}}\pi x\cdot \xi }\,\mathrm {d} \xi }" />.</center> </div> Remarque : cette formule dépend de la convention choisie pour la transformation de Fourier dans l'espace des fonctions. Elle est valide pour une <a href="#Conventions_alternatives">transformation de Fourier exprimée dans l'espace des fréquences</a>, dont la définition utilise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-{\rm {i}}2\pi \xi \cdot x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-{\rm {i}}2\pi \xi \cdot x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd596b4742f3997906df4d67659f6e516925d04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.89ex; height:2.676ex;" alt="{\displaystyle {\rm {e}}^{-{\rm {i}}2\pi \xi \cdot x}}" />. <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration de la formule d'inversion</div><div class="NavContent" style="padding-bottom:0.4em"> <ul><li>Prouvons d'abord que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" /> est stable par <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}}}" />. Par commodité, nous ne traiterons que le cas n = 1, mais le cas quelconque se traite de manière similaire. Soit donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {S}}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {S}}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3a94bbb23b9dcfa137c357ada80c9ac8256a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.099ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {S}}(\mathbb {R} )}" />.</li></ul> <ol><li>D'une part, la décroissance rapide implique que pour tout entier naturel n, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{n}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{n}f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a448de1d7c5c3a6a137d4e220f8252023d5ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.909ex; height:2.843ex;" alt="{\displaystyle x\mapsto x^{n}f(x)}" /> est intégrable. La fonction <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> <mi>f</mi> </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/a5dcde8694af884309e77803facff81e35836c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.205ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}f}" /> est donc définie et C∞.</li> <li>D'autre part, pour tout couple d'entiers naturels (n,k), la fonction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd6c1b194822cc3c6afb3c67126d87b3ec587bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.33ex; height:3.343ex;" alt="{\displaystyle [x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}}" /> est dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" />, donc dans L1. Sa transformée de Fourier tend vers 0 à l'infini. Or, en appliquant les propriétés d'échange entre multiplication par un polynôme et dérivation,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\left([x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}\right)(\xi )=(2{\rm {i}}\pi \xi )^{k}\left[{\mathcal {F}}\left(x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)\right)\right](\xi )=(2{\rm {i}}\pi \xi )^{k}[{\mathcal {F}}(f)]^{(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> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <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> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <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> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <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 {\mathcal {F}}\left([x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}\right)(\xi )=(2{\rm {i}}\pi \xi )^{k}\left[{\mathcal {F}}\left(x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)\right)\right](\xi )=(2{\rm {i}}\pi \xi )^{k}[{\mathcal {F}}(f)]^{(n)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86526fd390e231eb651f6b3ee93898e38085fccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:90.325ex; height:4.843ex;" alt="{\displaystyle {\mathcal {F}}\left([x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)]^{(k)}\right)(\xi )=(2{\rm {i}}\pi \xi )^{k}\left[{\mathcal {F}}\left(x\mapsto (-2{\rm {i}}\pi x)^{n}f(x)\right)\right](\xi )=(2{\rm {i}}\pi \xi )^{k}[{\mathcal {F}}(f)]^{(n)}(\xi )}" />,ce qui prouve la décroissance rapide de <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> <mi>f</mi> </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/a5dcde8694af884309e77803facff81e35836c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.205ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}f}" /> ainsi que toutes ses dérivées successives. Elle satisfait donc aux conditions d'appartenance à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" />.</li></ol> <ul><li>Soit f un élément de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" /> donc de L1. D'après le point précédent, <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> <mi>f</mi> </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/a5dcde8694af884309e77803facff81e35836c72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.205ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}f}" /> appartient aussi à L1. Le théorème d'inversion sur L1 s'applique et donne, en notant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow></mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95be1c8061ab7341c23ed3dde6b571c594baa09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.162ex; height:1.843ex;" alt="{\displaystyle {\tilde {}}}" /> l'opérateur de composition par –Id :<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\left(~{\tilde {}}\circ {\mathcal {F}}\circ {\mathcal {F}}\right)(f)=\left({\mathcal {F}}\circ {\tilde {}}\circ {\mathcal {F}}\right)(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow></mrow> <mo stretchy="false">~</mo> </mover> </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> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow></mrow> <mo stretchy="false">~</mo> </mover> </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> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\left(~{\tilde {}}\circ {\mathcal {F}}\circ {\mathcal {F}}\right)(f)=\left({\mathcal {F}}\circ {\tilde {}}\circ {\mathcal {F}}\right)(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e67e5a9a3044fc51332d5ead9bd17eb6a5f5eec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.435ex; height:2.843ex;" alt="{\displaystyle f=\left(~{\tilde {}}\circ {\mathcal {F}}\circ {\mathcal {F}}\right)(f)=\left({\mathcal {F}}\circ {\tilde {}}\circ {\mathcal {F}}\right)(f)}" />, ce qui prouve que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}:{\mathcal {S}}\to {\mathcal {S}}}"> <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> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}:{\mathcal {S}}\to {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3d859609636396e9937ed51cba87adf8c288a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.462ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}:{\mathcal {S}}\to {\mathcal {S}}}" /> est bijectif et que son inverse est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {}}\circ {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow></mrow> <mo stretchy="false">~</mo> </mover> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {}}\circ {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998a27b89f6909bf468f91e80c97c400c1848916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.284ex; height:2.176ex;" alt="{\displaystyle {\tilde {}}\circ {\mathcal {F}}}" />.</li></ul> </div><div class="clear" style="clear:both;"></div> </div> <div class="mw-heading mw-heading2"><h2 id="Transformation_de_Fourier_pour_les_distributions_tempérées">Transformation de Fourier pour les distributions tempérées</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=14" title="Modifier la section : Transformation de Fourier pour les distributions tempérées" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=14" title="Modifier le code source de la section : Transformation de Fourier pour les distributions tempérées">modifier le code</a>]</div> On définit la transformée de Fourier d'une <a href="/wiki/Distribution_temp%C3%A9r%C3%A9e" title="Distribution tempérée">distribution tempérée</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} ^{n})}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44614e5e873beb1e05f3d757bada9034fdc10b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.385ex; height:3.009ex;" alt="{\displaystyle T\in {\mathcal {S}}'(\mathbb {R} ^{n})}" /> comme la distribution définie via son crochet de dualité par <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \phi \in {\mathcal {S}}(\mathbb {R} ^{n})\quad \langle {\mathcal {F}}T,\phi \rangle =\langle T,{\mathcal {F}}\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> <mo fence="false" 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>T</mi> <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"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \phi \in {\mathcal {S}}(\mathbb {R} ^{n})\quad \langle {\mathcal {F}}T,\phi \rangle =\langle T,{\mathcal {F}}\phi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983cf87b63d6d9a42fe50b03d3cd2e7aea656396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.721ex; height:2.843ex;" alt="{\displaystyle \forall \phi \in {\mathcal {S}}(\mathbb {R} ^{n})\quad \langle {\mathcal {F}}T,\phi \rangle =\langle T,{\mathcal {F}}\phi \rangle }" />.</dd></dl> De même que sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" />, l'opérateur <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}}}" /> ainsi défini sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e577a88611d41be7e2cd935d34000ca976f9d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" /> est un automorphisme bicontinu. Les détails et des exemples ne sont pas donnés ici, mais figurent dans l'<a href="/wiki/Distribution_temp%C3%A9r%C3%A9e#Transformée_de_Fourier_des_distributions_tempérées" title="Distribution tempérée">article relatif aux distributions tempérées</a>. Remarquons que l'expression de la transformée de Fourier d'une fonction f ressemble au produit scalaire dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {L}}^{2}(\mathbb {C} ),(f,g)_{L^{2}}:=\int f{\bar {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <msub> <mo 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> <mo>∫</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {L}}^{2}(\mathbb {C} ),(f,g)_{L^{2}}:=\int f{\bar {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a88d2d075f9b9a72f414897646e4bc4f1f7609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.285ex; height:5.676ex;" alt="{\displaystyle {\rm {L}}^{2}(\mathbb {C} ),(f,g)_{L^{2}}:=\int f{\bar {g}}}" /> entre f et la conjuguée de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2\pi \xi }:x\mapsto {\rm {e}}^{{\rm {i}}2\pi \xi \cdot x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{2\pi \xi }:x\mapsto {\rm {e}}^{{\rm {i}}2\pi \xi \cdot x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d3b5094e44add713ce29e8688af83c6f5e3a72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.301ex; height:3.343ex;" alt="{\displaystyle e_{2\pi \xi }:x\mapsto {\rm {e}}^{{\rm {i}}2\pi \xi \cdot x}}" />. Sauf que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,e_{2\pi \xi })_{L^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,e_{2\pi \xi })_{L^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af231a2a38c0ab2c83356023d8e1eefcfca615b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.113ex; height:3.009ex;" alt="{\displaystyle (f,e_{2\pi \xi })_{L^{2}}}" /> n'a pas de sens car e2 π ξ n'est pas dans L2. Mais le crochet de dualité des distributions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle T_{f},e_{2\pi \xi }\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle T_{f},e_{2\pi \xi }\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0e0cf62e6c9676f9b5255884df2b955a988dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.145ex; height:3.009ex;" alt="{\displaystyle \langle T_{f},e_{2\pi \xi }\rangle }" />, qui pour les fonctions coïncide avec le <a href="/wiki/Produit_scalaire" title="Produit scalaire">produit scalaire</a> de L2, donne sens à cette formulation en tant que produit scalaire. Cette généralisation va bien plus loin car l'espace des distributions tempérées <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ca4a0bc72d2677618a39482625e3f749990455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}" /> englobe les différents objets sur lesquels la transformée de Fourier a été définie : fonctions de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <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> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" /> sommables ou de carré sommable, fonctions de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <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> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" /> périodiques localement sommables ou localement de carré sommable, suites discrètes sommables, suites discrètes périodiques. La transformée de Fourier sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ca4a0bc72d2677618a39482625e3f749990455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}" /> unifie et généralise les différentes définitions des transformées avec l'unique formalisme des distributions. Nous allons montrer que la transformée de Fourier sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e577a88611d41be7e2cd935d34000ca976f9d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" /> généralise les notions d'intégrales de Fourier et de séries de Fourier, en analysant successivement ces espaces. <div class="mw-heading mw-heading3"><h3 id="Compatibilités">Compatibilités</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=15" title="Modifier la section : Compatibilités" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=15" title="Modifier le code source de la section : Compatibilités">modifier le code</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Compatibilité_avec_les_espaces_de_fonctions">Compatibilité avec les espaces de fonctions</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=16" title="Modifier la section : Compatibilité avec les espaces de fonctions" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=16" title="Modifier le code source de la section : Compatibilité avec les espaces de fonctions">modifier le code</a>]</div> Les fonctions intégrables et les fonctions de carré sommable définissent des distributions tempérées. Montrons que les deux notions possibles de transformée de Fourier coïncident dans le cas L1, puis utilisons cette compatibilité pour l'établir dans le cas L2. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Compatibilité avec L1 et L2 — Soit <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f590ee38ad9f5c8219b14ed9c78934edb3d02e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.332ex; height:3.176ex;" alt="{\displaystyle f\in {\rm {L}}^{1}(\mathbb {R} ^{n})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}\in C_{0}(\mathbb {R} ^{n})}"> <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>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}\in C_{0}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e79ea7ed5a2d0a69297cdbe068fad8f05695a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.962ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}\in C_{0}(\mathbb {R} ^{n})}" /> sa transformée de Fourier,</li></ul> ou bien <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{2}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{2}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a9c285f2a248429b79d6fd40ec39bbff1cfa6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.332ex; height:3.176ex;" alt="{\displaystyle f\in {\rm {L}}^{2}(\mathbb {R} ^{n})}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}\in {\rm {L}}^{2}(\mathbb {R} ^{n})}"> <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>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}\in {\rm {L}}^{2}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f06ab70f997021728ffed2cdb5f8761fa99ffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.753ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}\in {\rm {L}}^{2}(\mathbb {R} ^{n})}" /> sa transformée de Fourier.</li></ul> Dans ces deux cas, <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}}}" /> définit une distribution tempérée égale à la transformée de Fourier de Tf, c'est-à-dire<center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}T_{f}=T_{\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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}T_{f}=T_{\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cee1879b563239224ab9c56d7a5cadb2464dc63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.424ex; height:3.343ex;" alt="{\displaystyle {\mathcal {F}}T_{f}=T_{\hat {f}}}" />.</center> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration</div><div class="NavContent" style="padding-bottom:0.4em"> <ul><li>Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb4d83b93e0181ded809577d113c74d3f6abba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.626ex; height:3.009ex;" alt="{\displaystyle f\in {\rm {L}}^{1}}" />. Il s'agit de vérifier que pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in {\mathcal {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \in {\mathcal {S}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d971de2c9f9229bf36982f98ceaf913ab3d71a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.718ex; height:2.509ex;" alt="{\displaystyle \phi \in {\mathcal {S}}}" />,</li></ul> <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\hat {\phi }}(x){\rm {d}}x=\int _{\mathbb {R} ^{n}}{\hat {f}}(y)\phi (y){\rm {d}}y}"> <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"> <mover> <mi>ϕ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\hat {\phi }}(x){\rm {d}}x=\int _{\mathbb {R} ^{n}}{\hat {f}}(y)\phi (y){\rm {d}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da7dd2dbf48b94970815c9172940ee318c91934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.333ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}f(x){\hat {\phi }}(x){\rm {d}}x=\int _{\mathbb {R} ^{n}}{\hat {f}}(y)\phi (y){\rm {d}}y}" />,</center> c'est-à-dire <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}f(x)\left(\int _{\mathbb {R} ^{n}}\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}y\right){\rm {d}}x=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}f(x){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}x\right)\phi (y){\rm {d}}y}"> <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> <mo>(</mo> <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>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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> <mo>(</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}f(x)\left(\int _{\mathbb {R} ^{n}}\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}y\right){\rm {d}}x=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}f(x){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}x\right)\phi (y){\rm {d}}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1cc5667bd1dd987895568e80a72431453d1d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.462ex; height:6.176ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}f(x)\left(\int _{\mathbb {R} ^{n}}\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}y\right){\rm {d}}x=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}f(x){\rm {e}}^{-{\rm {i}}x\cdot y}{\rm {d}}x\right)\phi (y){\rm {d}}y}" />.</center> Cela résulte simplement du <a href="/wiki/Th%C3%A9or%C3%A8me_de_Fubini" title="Théorème de Fubini">théorème de Fubini</a>, appliqué à la fonction intégrable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto f(x)\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto f(x)\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b77a07714f4d6143131dc0a210f483756c2f33c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.926ex; height:3.176ex;" alt="{\displaystyle (x,y)\mapsto f(x)\phi (y){\rm {e}}^{-{\rm {i}}x\cdot y}}" />. <ul><li>Les deux applications continues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto {\mathcal {F}}T_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto {\mathcal {F}}T_{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f499cbc4559feda017e24ba26f8c34fc0579dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.313ex; height:2.843ex;" alt="{\displaystyle f\mapsto {\mathcal {F}}T_{f}}" /> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto T_{\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto T_{\hat {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752ca3004d881f9a3586d7c381a457443520734e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.797ex; height:3.343ex;" alt="{\displaystyle f\mapsto T_{\hat {f}}}" />, de L2 dans l'<a href="/wiki/Espace_s%C3%A9par%C3%A9" title="Espace séparé">espace séparé</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e577a88611d41be7e2cd935d34000ca976f9d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" />, sont égales car elles coïncident, d'après le point précédent, sur le sous-espace dense L1 ∩ L2.</li></ul> </div><div class="clear" style="clear:both;"></div> </div> Enfin, les <a href="/wiki/Fonction_p%C3%A9riodique" title="Fonction périodique">fonctions périodiques</a> intégrables sur une période sont exactement les fonctions à la fois périodiques et localement intégrables, et donc définissent des distributions régulières. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Compatibilité avec L1per — La transformée de Fourier d'une distribution régulière Tf définie par une fonction T-périodique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\rm {L}}^{1}([0,T[)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">[</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\rm {L}}^{1}([0,T[)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0f09ad8952f593b86b7ab7e7d0f4b36213095a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.562ex; height:3.176ex;" alt="{\displaystyle f\in {\rm {L}}^{1}([0,T[)}" />, est la distribution à support discret correspondant à la suite de ses coefficients de Fourier : <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}T_{f}=\sum _{n\in \mathbb {Z} }c_{n}(f)\delta _{n}\quad {\rm {avec}}\quad c_{n}(f)=\int _{0}^{T}f(x){\rm {e}}^{-{\rm {i}}{\frac {2\pi }{T}}nx}\,\mathrm {d} 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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mspace width="1em"></mspace> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</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>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}T_{f}=\sum _{n\in \mathbb {Z} }c_{n}(f)\delta _{n}\quad {\rm {avec}}\quad c_{n}(f)=\int _{0}^{T}f(x){\rm {e}}^{-{\rm {i}}{\frac {2\pi }{T}}nx}\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788f68c05fb3818ca781be157f550726b6977007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.199ex; height:7.009ex;" alt="{\displaystyle {\mathcal {F}}T_{f}=\sum _{n\in \mathbb {Z} }c_{n}(f)\delta _{n}\quad {\rm {avec}}\quad c_{n}(f)=\int _{0}^{T}f(x){\rm {e}}^{-{\rm {i}}{\frac {2\pi }{T}}nx}\,\mathrm {d} x}" />.</center> </div> Le résultat énoncé ne concerne que les fonctions périodiques de la variable réelle mais s'étend facilement aux fonctions périodiques sur un réseau de ℝN. Comme la transformation de Fourier<a href="/wiki/Aide:Pr%C3%A9ciser_un_fait" title="Aide:Préciser un fait">[Laquelle ?]</a> possède une réciproque définie sur le même domaine, elle est de ce fait <a href="/wiki/Bijection" title="Bijection">bijective</a>, alors la démonstration de ce résultat sera une conséquence du théorème sur les distributions périodiques<a href="/wiki/Aide:Pr%C3%A9ciser_un_fait" title="Aide:Préciser un fait">[Lequel ?]</a>. <div class="mw-heading mw-heading4"><h4 id="Compatibilité_avec_les_espaces_de_suites">Compatibilité avec les espaces de suites</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=17" title="Modifier la section : Compatibilité avec les espaces de suites" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=17" title="Modifier le code source de la section : Compatibilité avec les espaces de suites">modifier le code</a>]</div> Les suites, c'est-à-dire les signaux discrets, peuvent parfois s'exprimer comme des distributions sur ℝ à support dans ℤ. À une suite donnée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60c817b80a08dbfde2304050ebc5e6567a377fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.644ex; height:2.843ex;" alt="{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}" /> correspond en effet de manière unique une série de masses de Dirac <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ac9f1036ebc0441e685e464b833b6fb9a4ae22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.386ex; height:5.676ex;" alt="{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}" />. Lorsque cette suite est sommable, cette série de <a href="/wiki/Masse_de_Dirac" class="mw-redirect" title="Masse de Dirac">masses de Dirac</a> a un sens en tant que distribution tempérée d'ordre 0. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Compatibilité de <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}}}" /> avec l1 — Soit une suite sommable à valeurs complexes notée <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60c817b80a08dbfde2304050ebc5e6567a377fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.644ex; height:2.843ex;" alt="{\displaystyle a:=(a_{n})_{n\in \mathbb {Z} }}" />. Sa transformée de Fourier à temps discret est une fonction 1-périodique qui coïncide avec la transformée de Fourier de la série de masses de Dirac associée à a. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {TFTD} [(a_{n})_{n\in \mathbb {Z} }]={\mathcal {F}}\left(\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">F</mi> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> <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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {TFTD} [(a_{n})_{n\in \mathbb {Z} }]={\mathcal {F}}\left(\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a735d0c284c27e3ece062744ae4dcd8f101a6145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.688ex; height:7.509ex;" alt="{\displaystyle \mathbf {TFTD} [(a_{n})_{n\in \mathbb {Z} }]={\mathcal {F}}\left(\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}\right)}" />.</dd></dl> </div> <div class="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration de la compatibilité de <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}}}" /> avec l1</div><div class="NavContent" style="padding-bottom:0.4em"> Lorsque a est sommable, la somme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ac9f1036ebc0441e685e464b833b6fb9a4ae22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.386ex; height:5.676ex;" alt="{\displaystyle T_{a}:=\sum _{k\in \mathbb {Z} }a_{k}\delta _{k}}" /> définit bien une distribution d'ordre 0. En effet, pour une fonction test <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in {\mathcal {S}}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \in {\mathcal {S}}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a733d034ad9da14b61207b8ebf9547a4c40a02a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.206ex; height:2.843ex;" alt="{\displaystyle \phi \in {\mathcal {S}}(\mathbb {R} )}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcl}\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle &=&\sum _{k\in \mathbb {Z} }(a_{k}\phi (k))\\\left|\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle \right|&\leq &\left(\sum _{k\in \mathbb {Z} }|a_{k}|\right)\|\phi \|_{\infty }.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">⟨</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mo>≤</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle &=&\sum _{k\in \mathbb {Z} }(a_{k}\phi (k))\\\left|\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle \right|&\leq &\left(\sum _{k\in \mathbb {Z} }|a_{k}|\right)\|\phi \|_{\infty }.\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfcd8e292da3c060d901627d9a332db34564b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.773ex; height:6.843ex;" alt="{\displaystyle {\begin{array}{lcl}\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle &=&\sum _{k\in \mathbb {Z} }(a_{k}\phi (k))\\\left|\langle \sum _{k\in \mathbb {Z} }a_{k}\delta _{k},\phi \rangle \right|&\leq &\left(\sum _{k\in \mathbb {Z} }|a_{k}|\right)\|\phi \|_{\infty }.\end{array}}}" /> Par continuité de la transformation de Fourier et formule de la transformée du Dirac <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\delta _{k})=e_{-2\pi k}}"> <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> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\delta _{k})=e_{-2\pi k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921df94374d32a98a61a468df089be33f83b1135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.17ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\delta _{k})=e_{-2\pi k}}" />, <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}T_{a}=\xi \mapsto \sum _{k\in \mathbb {Z} }a_{k}\mathrm {e} ^{-\mathrm {i} 2\pi k\xi }=\mathbf {TFTD} [a]}"> <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> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mi>ξ</mi> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>ξ</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">F</mi> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}T_{a}=\xi \mapsto \sum _{k\in \mathbb {Z} }a_{k}\mathrm {e} ^{-\mathrm {i} 2\pi k\xi }=\mathbf {TFTD} [a]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46717ba241d0266291b02766b70bb0a8a27ed55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.611ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}T_{a}=\xi \mapsto \sum _{k\in \mathbb {Z} }a_{k}\mathrm {e} ^{-\mathrm {i} 2\pi k\xi }=\mathbf {TFTD} [a]}" />.</dd></dl></dd></dl> On retrouve bien la transformée de Fourier en temps discret. </div><div class="clear" style="clear:both;"></div> </div> Par densité, la démonstration s'étend aux séries de carré sommable. Notons en outre que la transformation de Fourier des distributions périodiques donne une définition de la transformée de Fourier discrète de suites non nécessairement sommables : les suites à croissance polynomiale. En particulier, la <a href="/wiki/Transform%C3%A9e_de_Fourier_discr%C3%A8te" class="mw-redirect" title="Transformée de Fourier discrète">transformée de Fourier discrète</a> (TFD) s'interprète également comme la transformée d'une distribution tempérée. En effet, une suite finie de N points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace x_{k}\rbrace _{k=0}^{N-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lbrace x_{k}\rbrace _{k=0}^{N-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf23a02cc9b0b8ac38951bc8a365b8baca44dcde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.535ex; height:3.343ex;" alt="{\displaystyle \lbrace x_{k}\rbrace _{k=0}^{N-1}}" /> s'identifie de manière unique avec une suite N-périodique obtenue par périodisation, c'est-à-dire convolution avec un <a href="/wiki/Peigne_de_Dirac" title="Peigne de Dirac">peigne de Dirac</a>. <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid var(--border-color-base, #a2a9b1); text-align: justify;"> Compatibilité de <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}}}" /> avec la TFD — La TFD d'une suite x(•) à l'ordre N est la transformée de Fourier de la <a href="/wiki/Distribution_temp%C3%A9r%C3%A9e#Distributions_tempérées_à_support_dans_ℤN" title="Distribution tempérée">distribution à support dans ℤ</a> obtenue par périodisation de x(•) à la période N, c'est-à-dire convolution par un peigne de Dirac WN : <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {TFD} _{N}[x(.)](k)={\mathcal {F}}[{\tilde {x}}(.)](k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">F</mi> <mi mathvariant="bold">D</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 stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>k</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> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {TFD} _{N}[x(.)](k)={\mathcal {F}}[{\tilde {x}}(.)](k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72194891eb46295fd231eb457b2bf7252d8f3357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.282ex; height:2.843ex;" alt="{\displaystyle \mathbf {TFD} _{N}[x(.)](k)={\mathcal {F}}[{\tilde {x}}(.)](k)}" /> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}=W_{N}\ast \left(\sum _{n=0}^{N-1}x(n)\delta _{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>∗</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}=W_{N}\ast \left(\sum _{n=0}^{N-1}x(n)\delta _{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3cbde18eb2a2db4530176371cd29a6ee096448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.921ex; height:7.509ex;" alt="{\displaystyle {\tilde {x}}=W_{N}\ast \left(\sum _{n=0}^{N-1}x(n)\delta _{n}\right)}" />.</dd></dl></dd></dl> </div> <div class="mw-heading mw-heading3"><h3 id="Signaux_discrets_et_signaux_périodiques">Signaux discrets et signaux périodiques</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=18" title="Modifier la section : Signaux discrets et signaux périodiques" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=18" title="Modifier le code source de la section : Signaux discrets et signaux périodiques">modifier le code</a>]</div> Nous pouvons retenir que formellement, la transformée de Fourier échange discrétisation et périodisation. <ul><li>Le spectre d'un signal discret x[•] obtenu par échantillonnage à la période T présente un spectre périodique, résultant de la périodisation du spectre du signal continu :</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {TFTD} (x[.])={\mathcal {F}}(x(.))\ast W_{\frac {2\pi }{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">F</mi> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mo>.</mo> <mo stretchy="false">]</mo> <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>x</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∗</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {TFTD} (x[.])={\mathcal {F}}(x(.))\ast W_{\frac {2\pi }{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b02db3b48793d2e5e0612e594ded144763f785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.814ex; height:4.176ex;" alt="{\displaystyle \mathbf {TFTD} (x[.])={\mathcal {F}}(x(.))\ast W_{\frac {2\pi }{T}}}" />.</dd></dl> Si la multiplication n'est pas définie entre distribution, on donne dans le cas du peigne un sens à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[.]=x(.)\cdot W_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mo>.</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[.]=x(.)\cdot W_{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa46bed01ad6655db0da330f1d8cd6bd2d083ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.191ex; height:2.843ex;" alt="{\displaystyle x[.]=x(.)\cdot W_{T}}" />, et la formulation de convolution est encore vérifiée : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(x(.)\cdot W_{T})={\mathcal {F}}(x(.))\ast {\mathcal {F}}(W_{T})}"> <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>x</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <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>x</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <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> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(x(.)\cdot W_{T})={\mathcal {F}}(x(.))\ast {\mathcal {F}}(W_{T})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573b9d28a9192ff0cef536b963c1ba270e05282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.692ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(x(.)\cdot W_{T})={\mathcal {F}}(x(.))\ast {\mathcal {F}}(W_{T})}" />. <ul><li>Le spectre d'un signal T-périodique xT(•), c'est-à-dire la somme de sa série de Fourier, est celui obtenu par discrétisation du spectre du signal tronqué sur une seule période.</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(x_{T}(.))={\mathcal {F}}(x(.))\cdot W_{T}}"> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <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>x</mi> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(x_{T}(.))={\mathcal {F}}(x(.))\cdot W_{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec15a5a1e340b06788fa24d7c4b3c308e3c9b961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.567ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(x_{T}(.))={\mathcal {F}}(x(.))\cdot W_{T}}" /> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{T}.1_{[0,T]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <msub> <mn>.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{T}.1_{[0,T]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba863ab2ed739b41519e63232251eee4b2b2051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.54ex; height:3.009ex;" alt="{\displaystyle x=x_{T}.1_{[0,T]}}" />.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Liens_avec_d'autres_transformations">Liens avec d'autres transformations</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=19" title="Modifier la section : Liens avec d'autres transformations" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=19" title="Modifier le code source de la section : Liens avec d'autres transformations">modifier le code</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Lien_avec_les_transformations_de_Laplace">Lien avec les transformations de Laplace</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=20" title="Modifier la section : Lien avec les transformations de Laplace" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=20" title="Modifier le code source de la section : Lien avec les transformations de Laplace">modifier le code</a>]</div> La transformée de Fourier d'une fonction f est un cas particulier de la <a href="/wiki/Transform%C3%A9e_bilat%C3%A9rale_de_Laplace" class="mw-redirect" title="Transformée bilatérale de Laplace">transformée bilatérale de Laplace</a> de cette même fonction définie par : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{bil}\{f\}(p)=\int _{-\infty }^{+\infty }f(t)\,{\rm {e}}^{-pt}{\rm {d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>p</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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>p</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{bil}\{f\}(p)=\int _{-\infty }^{+\infty }f(t)\,{\rm {e}}^{-pt}{\rm {d}}t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92333ccf742e7cf0a33d94871b4c31b88849caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.695ex; height:6.176ex;" alt="{\displaystyle {\mathcal {L}}_{bil}\{f\}(p)=\int _{-\infty }^{+\infty }f(t)\,{\rm {e}}^{-pt}{\rm {d}}t}" /> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48f9350359615dc3385ac2aac5d8a14e114d74ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.778ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {C} }" />. On constate alors que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}_{bil}\{f\}({\rm {i}}\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 fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}_{bil}\{f\}({\rm {i}}\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b3b74f049aed80707068e885d8a8348b65ac23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.157ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}_{bil}\{f\}({\rm {i}}\xi )}" />. On peut également écrire ce lien en utilisant la <a href="/wiki/Transform%C3%A9e_de_Laplace" class="mw-redirect" title="Transformée de Laplace">transformée de Laplace</a> « usuelle » par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}\{f^{+}\}(+{\rm {i}}\xi )+{\mathcal {L}}\{f^{-}\}(-{\rm {i}}\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 fence="false" stretchy="false">{</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> <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">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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">L</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}\{f^{+}\}(+{\rm {i}}\xi )+{\mathcal {L}}\{f^{-}\}(-{\rm {i}}\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c7b472d0fc1862dcfb27684be077c879db9897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.417ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\{f\}(\xi )={\mathcal {L}}\{f^{+}\}(+{\rm {i}}\xi )+{\mathcal {L}}\{f^{-}\}(-{\rm {i}}\xi )}" /><a href="/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" title="Aide:Référence nécessaire">[réf. nécessaire]</a></dd></dl> où les fonctions f + et f – sont définies par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{+}(t)=f(+t),\ {\text{ si }}t\geq 0,0\ {\text{sinon}}.}"> <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>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> si </mtext> </mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>sinon</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{+}(t)=f(+t),\ {\text{ si }}t\geq 0,0\ {\text{sinon}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffc8fdd01e1bd6103d44294c19a0f37d53b539e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.488ex; height:3.009ex;" alt="{\displaystyle f^{+}(t)=f(+t),\ {\text{ si }}t\geq 0,0\ {\text{sinon}}.}" /></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-}(t)=f(-t),\ {\text{ si }}t\leq 0,0\ {\text{sinon}}.}"> <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>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> si </mtext> </mrow> <mi>t</mi> <mo>≤</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>sinon</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-}(t)=f(-t),\ {\text{ si }}t\leq 0,0\ {\text{sinon}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51211fc555d6f0df8405354e3e85ff43f0092a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.488ex; height:3.009ex;" alt="{\displaystyle f^{-}(t)=f(-t),\ {\text{ si }}t\leq 0,0\ {\text{sinon}}.}" /></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Lien_avec_les_séries_de_Fourier">Lien avec les séries de Fourier</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=21" title="Modifier la section : Lien avec les séries de Fourier" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=21" title="Modifier le code source de la section : Lien avec les séries de Fourier">modifier le code</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Parallèle_formel">Parallèle formel</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=22" title="Modifier la section : Parallèle formel" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=22" title="Modifier le code source de la section : Parallèle formel">modifier le code</a>]</div> La transformée de Fourier est définie de façon semblable : la variable d'intégration x est remplacée par nΔx, n étant l'indice de sommation, et l'intégrale par la somme. On a alors <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(k)=\Delta t\sum _{n=-\infty }^{\infty }f(n){\rm {e}}^{-{\rm {i}}2\pi kn\Delta t}}"> <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>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>t</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> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(k)=\Delta t\sum _{n=-\infty }^{\infty }f(n){\rm {e}}^{-{\rm {i}}2\pi kn\Delta t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e655ed517e8c545d0c9e8ab97c6783fbf656d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.606ex; height:6.843ex;" alt="{\displaystyle {\hat {f}}(k)=\Delta t\sum _{n=-\infty }^{\infty }f(n){\rm {e}}^{-{\rm {i}}2\pi kn\Delta t}}" />.</dd></dl> On trouvera quelques remarques à ce sujet dans <a href="/wiki/Analyse_spectrale" title="Analyse spectrale">Analyse spectrale</a>. <div class="mw-heading mw-heading4"><h4 id="Lien_direct">Lien direct</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=23" title="Modifier la section : Lien direct" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=23" title="Modifier le code source de la section : Lien direct">modifier le code</a>]</div> Cependant, comme indiqué par l'étude théorique dans la section précédente, un lien direct entre séries et transformées de Fourier est possible par la théorie des distributions. En reprenant de façon plus pratique l'exposé précédent, la transformée de Fourier (<a href="#Conventions_alternatives">définition fréquentielle</a>) d'une fonction périodique f de période T est un peigne de Dirac de période fréquentielle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{T}=1/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{T}=1/T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b970b0e631f23c8f1a2621a4ba5ad3e0d80fa46f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.597ex; height:2.843ex;" alt="{\displaystyle \nu _{T}=1/T}" />, modulé par des coefficients complexes cn : <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu )=\sum _{n=-\infty }^{+\infty }c_{n}\delta \left(\nu -n\nu _{T}\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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mi>ν</mi> <mo>−</mo> <mi>n</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu )=\sum _{n=-\infty }^{+\infty }c_{n}\delta \left(\nu -n\nu _{T}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc73e58f327ddc04942cdbaa2d28fa456aea52ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.922ex; height:7.176ex;" alt="{\displaystyle {\hat {f}}(\nu )=\sum _{n=-\infty }^{+\infty }c_{n}\delta \left(\nu -n\nu _{T}\right),}" /></center> où les cn sont précisément les coefficients de la <a href="/wiki/S%C3%A9rie_de_Fourier" title="Série de Fourier">série de Fourier</a> (complexe) de f. Pour le voir, il suffit de vérifier que la formule de transformation inverse de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu )}"> <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>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e41aa1f234052893a06bd98fd958d5f349d42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\nu )}" /> (définition en fréquences) donne précisément la série de Fourier de f, et donc qu'elle est égale à f presque partout (en supposant que la série de Fourier de f converge). Cela permet d'unifier le formalisme des séries de Fourier avec celui de la transformation de Fourier. Avec la <a href="#Conventions_alternatives">définition standard de la transformée de Fourier</a>, il faut remplacer la formule précédente par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\xi )=2\pi \sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\xi \over 2\pi }-{\frac {n}{T}}\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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>δ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ξ</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi )=2\pi \sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\xi \over 2\pi }-{\frac {n}{T}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04c92977774a88251d3f6e5e544614313ce5fc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.852ex; height:7.176ex;" alt="{\displaystyle {\hat {f}}(\xi )=2\pi \sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\xi \over 2\pi }-{\frac {n}{T}}\right).}" /></dd></dl> Avec la <a href="#Conventions_alternatives">définition pulsatoire</a>, et en notant la pulsation de <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}" /> par <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{T}=2\pi /T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{T}=2\pi /T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6062072ba21453e625e3540af4a81eafb6de1968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.227ex; height:2.843ex;" alt="{\displaystyle \omega _{T}=2\pi /T}" />, elle devient <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\omega )={\sqrt {2\pi }}\sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\omega -n\omega _{T} \over 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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> <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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>δ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ω</mi> <mo>−</mo> <mi>n</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\omega )={\sqrt {2\pi }}\sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\omega -n\omega _{T} \over 2\pi }\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f92b8615db385ac4cdb0a163b44a3197144c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.912ex; height:7.176ex;" alt="{\displaystyle {\hat {f}}(\omega )={\sqrt {2\pi }}\sum _{n=-\infty }^{+\infty }c_{n}\delta \left({\omega -n\omega _{T} \over 2\pi }\right).}" /></dd></dl> Par exemple, après quelques manipulations, on a les transformées de Fourier fréquentielles suivantes : <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {F}}e^{2i\pi \nu _{T}t}=\delta (\nu -\nu _{T})}"> <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> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>π</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {F}}e^{2i\pi \nu _{T}t}=\delta (\nu -\nu _{T})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca1738258a3694856b5e1f373a183a57b454ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.65ex; height:3.176ex;" alt="{\displaystyle {\cal {F}}e^{2i\pi \nu _{T}t}=\delta (\nu -\nu _{T})}" /> (Dirac décalé) ;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {F}}\cos(2\pi \nu _{T}t)={1 \over 2}\left[\delta (\nu -\nu _{T})+\delta (\nu +\nu _{T})\right]}"> <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>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {F}}\cos(2\pi \nu _{T}t)={1 \over 2}\left[\delta (\nu -\nu _{T})+\delta (\nu +\nu _{T})\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d763902f1243720780623a43a29e6fab7bbeca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.661ex; height:5.176ex;" alt="{\displaystyle {\cal {F}}\cos(2\pi \nu _{T}t)={1 \over 2}\left[\delta (\nu -\nu _{T})+\delta (\nu +\nu _{T})\right]}" /></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {F}}\sin(2\pi \nu _{T}t)={1 \over 2i}[\delta (\nu -\nu _{T})-\delta (\nu +\nu _{T})]}"> <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>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {F}}\sin(2\pi \nu _{T}t)={1 \over 2i}[\delta (\nu -\nu _{T})-\delta (\nu +\nu _{T})]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d712cbf03cc47c785f840d638f65e48d2082a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.821ex; height:5.176ex;" alt="{\displaystyle {\cal {F}}\sin(2\pi \nu _{T}t)={1 \over 2i}[\delta (\nu -\nu _{T})-\delta (\nu +\nu _{T})]}" /></li></ul> Il y a encore une formule utile qui donne les coefficients <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}}" /> de la série de Fourier d'une fonction périodique f dès que l'on connait la transformation de Fourier de sa « restriction » <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=1_{\tau }f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=1_{\tau }f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30cd5ef5695b718bce84f17889bc0073f52bbcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.738ex; height:2.509ex;" alt="{\displaystyle g=1_{\tau }f}" /> à la période centrale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =[-T/2,\ T/2]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =[-T/2,\ T/2]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d554736f86501ce57b1dd2a32ff148a72d3bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.939ex; height:2.843ex;" alt="{\displaystyle \tau =[-T/2,\ T/2]}" /> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9351259e90225fb86e6d884862bd7ccc579db8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\displaystyle {\hat {g}}}" /> existe nécessairement si f est <a href="/wiki/Fonction_localement_int%C3%A9grable" title="Fonction localement intégrable">localement intégrable</a> puisque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /> est compacte). En effet, par comparaison de la formule des <a href="/wiki/S%C3%A9rie_de_Fourier#Principe_des_séries_de_Fourier" title="Série de Fourier">coefficients</a> de la série de Fourier de <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}" /> avec celle donnant la transformée de Fourier inverse de <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/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" />, on obtient facilement, pour la définition fréquentielle, que <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={\frac {1}{T}}{\hat {g}}(n\nu _{T})=\nu _{T}{\hat {g}}(n\nu _{T}).}"> <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>T</mi> </mfrac> </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>n</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <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>n</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={\frac {1}{T}}{\hat {g}}(n\nu _{T})=\nu _{T}{\hat {g}}(n\nu _{T}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c60d20f6965cb5c4c2919322f51de60fb6aa02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.026ex; height:5.176ex;" alt="{\displaystyle c_{n}={\frac {1}{T}}{\hat {g}}(n\nu _{T})=\nu _{T}{\hat {g}}(n\nu _{T}).}" /></center> Pour la définition standard de la transformée de Fourier, cette formule devient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={1 \over T}{\hat {g}}(n\omega _{T}),}"> <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>T</mi> </mfrac> </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>n</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={1 \over T}{\hat {g}}(n\omega _{T}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f4beeef502ddb84ed87c60d29299b522005f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.714ex; height:5.176ex;" alt="{\displaystyle c_{n}={1 \over T}{\hat {g}}(n\omega _{T}),}" /> avec <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{T}=2\pi /T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{T}=2\pi /T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6062072ba21453e625e3540af4a81eafb6de1968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.227ex; height:2.843ex;" alt="{\displaystyle \omega _{T}=2\pi /T}" />, et pour la définition pulsatoire, elle devient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={1 \over {\sqrt {2\pi }}}\omega _{T}{\hat {g}}(n\omega _{T}).}"> <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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <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>n</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={1 \over {\sqrt {2\pi }}}\omega _{T}{\hat {g}}(n\omega _{T}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c69eeeaae6929912fa69b33ab0b21e4a78dff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.343ex; height:6.176ex;" alt="{\displaystyle c_{n}={1 \over {\sqrt {2\pi }}}\omega _{T}{\hat {g}}(n\omega _{T}).}" /> Cette formule permet l'utilisation de l'imposante machinerie disponible pour la transformation de Fourier (convolution, décalage, produit, distributions, tables, etc.) pour le calcul des coefficients de Fourier d'une fonction périodique. On peut ainsi facilement obtenir la série de Fourier de trains d'ondes pulsées de forme carrée, triangulaire, demi-sinusoïdale, etc. Par exemple, quelle est la série de Fourier correspondant à un train de pulses étroits, de masse 1 et de période T grande relativement à la durée des pulses ? Approximons chaque pulse par un Dirac <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 }" />. La transformée de Fourier fréquentielle de <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 }" /> est la fonction identiquement égale à 1 (voir table ci-dessous). Donc la formule précédente donne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}={1 \over T}.}"> <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>T</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}={1 \over T}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18e757f2c72f6f1fa81156f703256080b74ba6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.443ex; height:5.176ex;" alt="{\displaystyle c_{n}={1 \over T}.}" /> Ainsi la série de Fourier du train de pulses est <center><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{n=-\infty }^{+\infty }{1 \over T}{\rm {e}}^{{\rm {i}}2\pi {\tfrac {n}{T}}x}={1 \over T}+\sum _{n=1}^{\infty }{2 \over T}\cos(2\pi {\tfrac {n}{T}}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> <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"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>T</mi> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>T</mi> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>T</mi> </mfrac> </mstyle> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=-\infty }^{+\infty }{1 \over T}{\rm {e}}^{{\rm {i}}2\pi {\tfrac {n}{T}}x}={1 \over T}+\sum _{n=1}^{\infty }{2 \over T}\cos(2\pi {\tfrac {n}{T}}x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f126fe9742c293d85af9f918749160ba616adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.732ex; height:7.176ex;" alt="{\displaystyle f(x)=\sum _{n=-\infty }^{+\infty }{1 \over T}{\rm {e}}^{{\rm {i}}2\pi {\tfrac {n}{T}}x}={1 \over T}+\sum _{n=1}^{\infty }{2 \over T}\cos(2\pi {\tfrac {n}{T}}x)}" /></center> (au sens des distributions). <div class="mw-heading mw-heading4"><h4 id="Autre_interprétation">Autre interprétation</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=24" title="Modifier la section : Autre interprétation" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=24" title="Modifier le code source de la section : Autre interprétation">modifier le code</a>]</div> <a href="#Transformation_de_Fourier_inverse">Comme on l'a vu plus haut</a>, il est d'autre part possible d'interpréter l'intégrale de la transformée de Fourier comme une somme finie de n <a href="/wiki/Oscillateur_harmonique" title="Oscillateur harmonique">oscillateurs harmoniques</a>, où n est un <a href="/wiki/Analyse_non-standard" class="mw-redirect" title="Analyse non-standard">entier non standard</a><a href="#cite_note-5">[5]</a> ; cela revient à identifier (en un sens différent) la transformation de Fourier aux coefficients d'une série de Fourier. <div class="mw-heading mw-heading4"><h4 id="Transformée">Transformée</h4>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=25" title="Modifier la section : Transformée" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=25" title="Modifier le code source de la section : Transformée">modifier le code</a>]</div> On utilise les variables normalisées suivantes : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={f \over f_{e}}=f\Delta t=f|_{\Delta t=1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={f \over f_{e}}=f\Delta t=f|_{\Delta t=1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640c9ca3f128f4e6999b8a197992a2d2c4272ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.285ex; height:5.843ex;" alt="{\displaystyle F={f \over f_{e}}=f\Delta t=f|_{\Delta t=1}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =2\pi F=2\pi f\Delta t=\omega \Delta t=\omega |_{\Delta t=1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>F</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mi>ω</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mi>ω</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =2\pi F=2\pi f\Delta t=\omega \Delta t=\omega |_{\Delta t=1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26045d0c9c2927672f1153be1dc5476bb54a5410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.465ex; height:3.009ex;" alt="{\displaystyle \Omega =2\pi F=2\pi f\Delta t=\omega \Delta t=\omega |_{\Delta t=1}}" />. <table class="wikitable"> <tbody><tr> <th>Transformation de Fourier (analyse)</th> <th>Transformation inverse (synthèse) </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(f)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi fn\Delta t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>t</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> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(f)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi fn\Delta t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6db2e8577208e02b073bdc5119d46139f272ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.053ex; height:6.843ex;" alt="{\displaystyle X(f)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi fn\Delta t}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(n)=\int _{f_{e}}X(f){\rm {e}}^{{\rm {i}}2\pi fn\Delta t}{\rm {d}}f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(n)=\int _{f_{e}}X(f){\rm {e}}^{{\rm {i}}2\pi fn\Delta t}{\rm {d}}f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac11be8921e3346ce9d368b7253f4a5e7241aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.114ex; height:6.176ex;" alt="{\displaystyle x(n)=\int _{f_{e}}X(f){\rm {e}}^{{\rm {i}}2\pi fn\Delta t}{\rm {d}}f}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(w)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}\omega n\Delta t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>t</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> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(w)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}\omega n\Delta t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a391fd3b8a1e7c58bcb545c6724f8aec13b89b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.793ex; height:6.843ex;" alt="{\displaystyle X(w)=\Delta t\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}\omega n\Delta t}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(n)={1 \over 2\pi }\int _{\omega _{2}=2\pi f_{e}}X(w){\rm {e}}^{{\rm {i}}wn\Delta t}{\rm {d}}w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mi>X</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>w</mi> <mi>n</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(n)={1 \over 2\pi }\int _{\omega _{2}=2\pi f_{e}}X(w){\rm {e}}^{{\rm {i}}wn\Delta t}{\rm {d}}w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab99860a38ea01a0e3b8641c9210d6b19e1cde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.007ex; height:6.176ex;" alt="{\displaystyle x(n)={1 \over 2\pi }\int _{\omega _{2}=2\pi f_{e}}X(w){\rm {e}}^{{\rm {i}}wn\Delta t}{\rm {d}}w}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(F)=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi nF}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>F</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>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>F</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(F)=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi nF}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cece65766f3eb6efb13e94b8d3bf1d2a433fa74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.717ex; height:6.843ex;" alt="{\displaystyle X(F)=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}2\pi nF}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(n)=\int _{1}X(f){\rm {e}}^{{\rm {i}}2\pi nF}{\rm {d}}F\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>F</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>F</mi> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(n)=\int _{1}X(f){\rm {e}}^{{\rm {i}}2\pi nF}{\rm {d}}F\,\!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b3dfdce93b78532784636542b2bf33198d0d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:24.557ex; height:5.676ex;" alt="{\displaystyle x(n)=\int _{1}X(f){\rm {e}}^{{\rm {i}}2\pi nF}{\rm {d}}F\,\!}" /> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(\Omega )=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}n\Omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</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>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>n</mi> <mi mathvariant="normal">Ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(\Omega )=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}n\Omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f443efb594a4540497aaf323e738ce2c544113a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.846ex; height:6.843ex;" alt="{\displaystyle X(\Omega )=\sum _{n=-\infty }^{\infty }x(n){\rm {e}}^{-{\rm {i}}n\Omega }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(n)={1 \over 2\pi }\int _{2\pi }X(\Omega ){\rm {e}}^{{\rm {i}}n\Omega }{\rm {d}}\Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msub> <mi>X</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>n</mi> <mi mathvariant="normal">Ω</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(n)={1 \over 2\pi }\int _{2\pi }X(\Omega ){\rm {e}}^{{\rm {i}}n\Omega }{\rm {d}}\Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9d554d8d54edfa08f66a1fe3e9322f61c3e1d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.358ex; height:5.676ex;" alt="{\displaystyle x(n)={1 \over 2\pi }\int _{2\pi }X(\Omega ){\rm {e}}^{{\rm {i}}n\Omega }{\rm {d}}\Omega }" /> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Généralisation">Généralisation</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=26" title="Modifier la section : Généralisation" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=26" title="Modifier le code source de la section : Généralisation">modifier le code</a>]</div> La transformée de Fourier se généralise pratiquement telle quelle aux <a href="/wiki/Groupe_ab%C3%A9lien" title="Groupe abélien">groupes abéliens</a> localement compacts, grâce à la <a href="/wiki/Dualit%C3%A9_de_Pontryagin" class="mw-redirect" title="Dualité de Pontryagin">dualité de Pontryagin</a>. En <a href="/wiki/Traitement_d%27images" title="Traitement d'images">traitement d'images</a>, on effectue des transformations de Fourier à deux dimensions : si f est une fonction de ℝ2 dans ℝ, sa transformée de Fourier est définie par : <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(u,v)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\mathrm {e} ^{-{\rm {i}}(ux+vy)}\,\mathrm {d} x\mathrm {d} y}"> <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>u</mi> <mo>,</mo> <mi>v</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> <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>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(u,v)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\mathrm {e} ^{-{\rm {i}}(ux+vy)}\,\mathrm {d} x\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48cc9621e8b13cb1015118f76da8174844007d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.883ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(u,v)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\mathrm {e} ^{-{\rm {i}}(ux+vy)}\,\mathrm {d} x\mathrm {d} y}" />.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Tables_des_principales_transformées_de_Fourier">Tables des principales transformées de Fourier</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=27" title="Modifier la section : Tables des principales transformées de Fourier" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=27" title="Modifier le code source de la section : Tables des principales transformées de Fourier">modifier le code</a>]</div> Les tableaux suivants présentent les transformations de Fourier de certaines fonctions. Les transformées de Fourier de f (x), g(x) et h(x) sont notées respectivement f̂, ĝ et ĥ. N'apparaissent que les trois conventions les plus courantes. Il peut être utile de noter que l'entrée sur la dualité indique une relation entre la transformée de Fourier d'une fonction et la fonction d'origine, ce qui peut être considéré comme une relation entre la transformation de Fourier et son inverse. <div class="mw-heading mw-heading3"><h3 id="Relations_entre_fonctions_à_une_variable">Relations entre fonctions à une variable</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=28" title="Modifier la section : Relations entre fonctions à une variable" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=28" title="Modifier le code source de la section : Relations entre fonctions à une variable">modifier le code</a>]</div> Les transformées de Fourier de ce tableau sont traitées dans <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Arthur Erdélyi, <cite class="italique" lang="en">Tables of Integral Transforms, Vol. 1</cite>, McGraw-Hill, <time>1954</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tables+of+Integral+Transforms%2C+Vol.+1&rft.pub=McGraw-Hill&rft.aulast=Erd%C3%A9lyi&rft.aufirst=Arthur&rft.date=1954&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier"> ou <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> David Kammler, <cite class="italique" lang="en">A First Course in Fourier Analysis</cite>, USA, Prentice Hall, <time>2000</time><span class="Z3988" 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.place=USA&rft.pub=Prentice+Hall&rft.aulast=Kammler&rft.aufirst=David&rft.date=2000&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">. <table class="wikitable"> <tbody><tr> <th>Fonction</th> <th>Transformée de Fourier ξ est la fréquence </th> <th>Transformée de Fourier ω = 2πξ est la pulsation ou fréquence angulaire</th> <th>Transformée de Fourier définition alternative </th> <th>Remarques </th></tr> <tr> <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"></mspace> </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 )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,\mathrm {d} 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> <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> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9192ad0bdff62aaed96189972d43310a6475dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:22.543ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,\mathrm {d} x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} 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> <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> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b854b4e123327c9e2c853458df5a85ac54b2ad90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.727ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} 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> <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>ν</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f174616e75f34407c919c10500bc9f117a72e69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:20.922ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} x\end{aligned}}}" /> </td> <td>Définition </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot f(x)+b\cdot g(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot f(x)+b\cdot g(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3a0f6a96407a91ae2dbfa8ca0d682bb815e67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.486ex; height:2.843ex;" alt="{\displaystyle a\cdot f(x)+b\cdot g(x)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <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> <mo>+</mo> <mi>b</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 stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090f7b56ba13574bfbe1e4d2b36e77e9453eef3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.422ex; height:3.343ex;" alt="{\displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <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>ω</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</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 stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5529f9e13b9f913c26b78ae143034868fcbc57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:3.343ex;" alt="{\displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <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>ν</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</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 stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f622e22a7cff33438e227829e9a3966ba67f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.827ex; height:3.343ex;" alt="{\displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,}" /> </td> <td>Linéarité </td></tr> <tr> <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"></mspace> </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 {\rm {e}}^{-2\pi {\rm {i}}a\xi }{\hat {f}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>ξ</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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-2\pi {\rm {i}}a\xi }{\hat {f}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4033d08cdb016e7997f1a9b8d4f66865d3e76073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.288ex; height:3.343ex;" alt="{\displaystyle {\rm {e}}^{-2\pi {\rm {i}}a\xi }{\hat {f}}(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-{\rm {i}}a\omega }{\hat {f}}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>ω</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>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-{\rm {i}}a\omega }{\hat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/305b86ef8f31a79b341f7e6a450a2a53351eb758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.234ex; height:3.343ex;" alt="{\displaystyle {\rm {e}}^{-{\rm {i}}a\omega }{\hat {f}}(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-{\rm {i}}a\nu }{\hat {f}}(\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>ν</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>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-{\rm {i}}a\nu }{\hat {f}}(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ea1a96e1489e5a911c42804c1318ab860bee45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.869ex; height:3.343ex;" alt="{\displaystyle {\rm {e}}^{-{\rm {i}}a\nu }{\hat {f}}(\nu )\,}" /> </td> <td>Décalage dans le temps </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x){\rm {e}}^{{\rm {i}}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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x){\rm {e}}^{{\rm {i}}ax}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63edd3181f042d4665f07c0799bf72f35938800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.337ex; height:3.176ex;" alt="{\displaystyle f(x){\rm {e}}^{{\rm {i}}ax}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {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 stretchy="false">^</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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637f31f6ea26d7a07ee3aa5f0ffc5eef9c152d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.45ex; height:4.843ex;" alt="{\displaystyle {\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\omega -a)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9b6c0a046c781e88874143614a8f2dadf9d31c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.411ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\omega -a)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu -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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu -a)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2764dd25c759d1a5a151520bc154889e11a159ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.198ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\nu -a)\,}" /> </td> <td>Décalage dans le domaine des fréquences, relation duale de la formule du décalage dans le temps </td></tr> <tr> <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"></mspace> </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|}}{\hat {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 stretchy="false">^</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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2da629ed50ce63167dbd338c365cad5611a66b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.32ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,}" /> </td> <td><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"></mspace> </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)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{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>ν</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782f0646cdfb0029b6d08e4983ed6ad1566c53e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.677ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,}" /> </td> <td>Changement d'échelle des temps. Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61d5baa05004815f3abc52f517ce62b609b9b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.523ex; height:2.843ex;" alt="{\displaystyle |a|}" /> est grand, alors f (ax) est resserré autour de 0 et <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"></mspace> </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)\,}" /> s'étale et s’aplatit. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(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> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7633eedd72432dc87b463c9cdedd717e680b1ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.225ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(x)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20e05a2589b258e2b2386566b2edcb01be731c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.313ex; height:2.843ex;" alt="{\displaystyle f(-\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00bf6bf9adeb642bc3d55f0bdefe4a3d94b6e5be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.729ex; height:2.843ex;" alt="{\displaystyle f(-\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi f(-\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi f(-\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d564c3c6b932b1d909cb059fd097df482484169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.01ex; height:2.843ex;" alt="{\displaystyle 2\pi f(-\nu )\,}" /> </td> <td>Dualité. Ici f̂ doit être calculée en utilisant la même formule que dans la colonne transformation de Fourier. Cela résulte d'un changement de la variable "muette", de x à ξ ou ω ou ν. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{n}f(x)}{\mathrm {d} x^{n}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{n}f(x)}{\mathrm {d} x^{n}}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43519528c5d73a96bc252460e977c0bec6eb8cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.152ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{n}f(x)}{\mathrm {d} x^{n}}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2\pi {\rm {i}}\xi )^{n}{\hat {f}}(\xi )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2\pi {\rm {i}}\xi )^{n}{\hat {f}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c54e40dab063ba170aeb9c2a8bcb4dd236e8a8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.125ex; height:3.343ex;" alt="{\displaystyle (2\pi {\rm {i}}\xi )^{n}{\hat {f}}(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\rm {i}}\omega )^{n}{\hat {f}}(\omega )\,}"> <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 mathvariant="normal">i</mi> </mrow> </mrow> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {i}}\omega )^{n}{\hat {f}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf05acf1129448e93fbb09e0a6f0b30763a55b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.462ex; height:3.343ex;" alt="{\displaystyle ({\rm {i}}\omega )^{n}{\hat {f}}(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\rm {i}}\nu )^{n}{\hat {f}}(\nu )\,}"> <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 mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</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>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {i}}\nu )^{n}{\hat {f}}(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9791e3ce98f40f33cf4037e02e2525247b060a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.034ex; height:3.343ex;" alt="{\displaystyle ({\rm {i}}\nu )^{n}{\hat {f}}(\nu )\,}" /> </td> <td> </td></tr> <tr> <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"></mspace> </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 {\rm {i}}{2\pi }}\right)^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\xi )}{\mathrm {d} \xi ^{n}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\rm {i}}{2\pi }}\right)^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\xi )}{\mathrm {d} \xi ^{n}}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc292fad94363673ddd9d0653bfa2eb7cd2fad38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.243ex; height:6.843ex;" alt="{\displaystyle \left({\frac {\rm {i}}{2\pi }}\right)^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\xi )}{\mathrm {d} \xi ^{n}}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\omega )}{\mathrm {d} \omega ^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <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>ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <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 {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\omega )}{\mathrm {d} \omega ^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ec1ceb7bcbec0ba38359b4b4cb90ead3fe5c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.167ex; height:6.343ex;" alt="{\displaystyle {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\omega )}{\mathrm {d} \omega ^{n}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\nu )}{\mathrm {d} \nu ^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <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>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\nu )}{\mathrm {d} \nu ^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a63a6715f4642b3c1abbbdf232869c862255d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.953ex; height:6.343ex;" alt="{\displaystyle {\rm {i}}^{n}{\frac {\mathrm {d} ^{n}{\hat {f}}(\nu )}{\mathrm {d} \nu ^{n}}}}" /> </td> <td>Relation duale de la précédente </td></tr> <tr> <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"></mspace> </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 {\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>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"> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f428b7948213fa1d57c736c69a69fa9ecedf0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.996ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {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> <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> <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>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b104d6bf303cf7efedb3d0fe5b0760a605b6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.258ex; height:3.343ex;" alt="{\displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu ){\hat {g}}(\nu )\,}"> <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>ν</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>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu ){\hat {g}}(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b17d09bd25aba0f1590c518ec59604a090dff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:3.343ex;" alt="{\displaystyle {\hat {f}}(\nu ){\hat {g}}(\nu )\,}" /> </td> <td>La notation f ∗ g signifie le <a href="/wiki/Produit_de_convolution" title="Produit de convolution">produit de convolution</a> de f et g — cette règle est le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Fubini" title="Théorème de Fubini">théorème de Fubini</a> </td></tr> <tr> <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"></mspace> </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({\hat {f}}*{\hat {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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\hat {f}}*{\hat {g}}\right)(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56315238a0ddc615eb8ffa3a9e16e0ca655ef7c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.514ex; height:4.843ex;" alt="{\displaystyle \left({\hat {f}}*{\hat {g}}\right)(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\hat {f}}*{\hat {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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\hat {f}}*{\hat {g}}\right)(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd006d9dbceda28279e24f2ad5c7b50b36e59b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.584ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi }}}\left({\hat {f}}*{\hat {g}}\right)(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2\pi }}\left({\hat {f}}*{\hat {g}}\right)(\nu )\,}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi }}\left({\hat {f}}*{\hat {g}}\right)(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd366e45af1922b7726954698787f6f96c161e25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.434ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2\pi }}\left({\hat {f}}*{\hat {g}}\right)(\nu )\,}" /> </td> <td>Relation duale du théorème de Fubini </td></tr> <tr> <td>Si f (x) est purement réelle </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\xi )={\overline {{\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> <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 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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a96750879daa1ba8936d36759d6e3254726ec38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.485ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {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 stretchy="false">^</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 stretchy="false">^</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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1df7ac8c1261587b94cda44b659557d9264f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.317ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,}"> <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> <mo>−</mo> <mi>ν</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>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4cd3faea8b5256abe86312a93cae09a2c80211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.89ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,}" /> </td> <td>Symétrie hermitienne. z est la notation du <a href="/wiki/Complexe_conjugu%C3%A9" class="mw-redirect" title="Complexe conjugué">complexe conjugué</a> de z. </td></tr> <tr> <td>Si f (x) est purement réelle et <a href="/wiki/Parit%C3%A9_d%27une_fonction" title="Parité d'une fonction">paire</a> </td> <td colspan="4" align="center">f̂ (ξ), f̂ (ω) et f̂ (ν) sont purement réelles et <a href="/wiki/Fonction_paire" class="mw-redirect" title="Fonction paire">paires</a>. </td></tr> <tr> <td>Si f (x) est purement réelle et <a href="/wiki/Parit%C3%A9_d%27une_fonction" title="Parité d'une fonction">impaire</a> </td> <td colspan="4" align="center">f̂ (ξ), f̂ (ω) et f̂ (ν) sont purement <a href="/wiki/Nombre_imaginaire_pur" title="Nombre imaginaire pur">imaginaires</a> et <a href="/wiki/Parit%C3%A9_d%27une_fonction" title="Parité d'une fonction">impaires</a>. </td></tr> <tr> <td>Si f (x) est imaginaire pur </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\xi )=-{\overline {{\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> <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 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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\xi )=-{\overline {{\hat {f}}(\xi )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7883bc07aa87599da9fff804a5aa1fea5863e5a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.294ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\xi )=-{\overline {{\hat {f}}(\xi )}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\omega )=-{\overline {{\hat {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 stretchy="false">^</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 stretchy="false">^</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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\omega )=-{\overline {{\hat {f}}(\omega )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1738115076a10dd463798be1ab961119938053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.125ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\omega )=-{\overline {{\hat {f}}(\omega )}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(-\nu )=-{\overline {{\hat {f}}(\nu )}}\,}"> <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> <mo>−</mo> <mi>ν</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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(-\nu )=-{\overline {{\hat {f}}(\nu )}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc5aaae32602709a13678a3790c0d7ec450a8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.698ex; height:4.176ex;" alt="{\displaystyle {\hat {f}}(-\nu )=-{\overline {{\hat {f}}(\nu )}}\,}" /> </td> <td>z est le <a href="/wiki/Complexe_conjugu%C3%A9" class="mw-redirect" title="Complexe conjugué">complexe conjugué</a> de z. </td></tr> <tr> <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 {{\hat {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 stretchy="false">^</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 {{\hat {f}}(-\xi )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52d6937ba549b3d9d6405886acba38b65b117e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:4.176ex;" alt="{\displaystyle {\overline {{\hat {f}}(-\xi )}}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\hat {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 stretchy="false">^</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 {{\hat {f}}(-\omega )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae78a7fc1b8497c94414ee72362d56e243786e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:4.176ex;" alt="{\displaystyle {\overline {{\hat {f}}(-\omega )}}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {{\hat {f}}(-\nu )}}}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {{\hat {f}}(-\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0c1eb98637de54ed933a3ae1e198b36dfbba97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.664ex; height:4.176ex;" alt="{\displaystyle {\overline {{\hat {f}}(-\nu )}}}" /></td> <td><a href="/wiki/Conjugu%C3%A9" title="Conjugué">Conjugaison complexe</a> </td></tr> <tr> <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 {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\hat {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 stretchy="false">^</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 stretchy="false">^</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 {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f52aa7b1e542a7cf0612dd61551d98eef70fd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.341ex; height:7.509ex;" alt="{\displaystyle {\frac {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)+{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\hat {f}}(\omega -a)+{\hat {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 stretchy="false">^</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 stretchy="false">^</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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\hat {f}}(\omega -a)+{\hat {f}}(\omega +a)}{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5617a02765c6f79c5de0ad07072ab9bfe3f0493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.112ex; height:6.176ex;" alt="{\displaystyle {\frac {{\hat {f}}(\omega -a)+{\hat {f}}(\omega +a)}{2}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\hat {f}}(\nu -a)+{\hat {f}}(\nu +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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</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 {{\hat {f}}(\nu -a)+{\hat {f}}(\nu +a)}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46bb21cf2151504e7deac7149aec0ee55e6b6b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.298ex; height:6.176ex;" alt="{\displaystyle {\frac {{\hat {f}}(\nu -a)+{\hat {f}}(\nu +a)}{2}}}" /> </td> <td>Peut se déduire de la <a href="/wiki/Formule_d%27Euler" title="Formule d'Euler">formule d'Euler</a> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}+{\rm {e}}^{-{\rm {i}}ax}}{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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}+{\rm {e}}^{-{\rm {i}}ax}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcf6346b081b417b592ac1a13cee5a74eea09b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.597ex; height:5.676ex;" alt="{\displaystyle \cos(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}+{\rm {e}}^{-{\rm {i}}ax}}{2}}}" /> </td></tr> <tr> <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 {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2{\rm {i}}}}}"> <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 stretchy="false">^</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 stretchy="false">^</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2{\rm {i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ecc08a6e14199e246abb7cb5c872ac35a54300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.341ex; height:7.509ex;" alt="{\displaystyle {\frac {{\hat {f}}\left(\xi -{\frac {a}{2\pi }}\right)-{\hat {f}}\left(\xi +{\frac {a}{2\pi }}\right)}{2{\rm {i}}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2{\rm {i}}}}}"> <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 stretchy="false">^</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 stretchy="false">^</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2{\rm {i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b07ffc022037ab31628a7c5a306f5725f2be8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.725ex; height:6.176ex;" alt="{\displaystyle {\frac {{\hat {f}}(\omega -a)-{\hat {f}}(\omega +a)}{2{\rm {i}}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\hat {f}}(\nu -a)-{\hat {f}}(\nu +a)}{2{\rm {i}}}}}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\hat {f}}(\nu -a)-{\hat {f}}(\nu +a)}{2{\rm {i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0ba3dd48ae9c5c1b925df10f3723f98f8475cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.298ex; height:6.176ex;" alt="{\displaystyle {\frac {{\hat {f}}(\nu -a)-{\hat {f}}(\nu +a)}{2{\rm {i}}}}}" /> </td> <td>Peut se déduire de la <a href="/wiki/Formule_d%27Euler" title="Formule d'Euler">formule d'Euler</a> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}-{\rm {e}}^{-{\rm {i}}ax}}{2{\rm {i}}}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}-{\rm {e}}^{-{\rm {i}}ax}}{2{\rm {i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d94949d72eef8ea6f6d45f15c46f8a1133bd4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.342ex; height:5.676ex;" alt="{\displaystyle \sin(ax)={\frac {{\rm {e}}^{{\rm {i}}ax}-{\rm {e}}^{-{\rm {i}}ax}}{2{\rm {i}}}}}" /> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Fonctions_de_carré_intégrable_à_une_variable">Fonctions de <a href="/wiki/Carr%C3%A9_sommable" title="Carré sommable">carré intégrable</a> à une variable</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=29" title="Modifier la section : Fonctions de carré intégrable à une variable" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=29" title="Modifier le code source de la section : Fonctions de carré intégrable à une variable">modifier le code</a>]</div> Les transformées de Fourier de ce tableau peuvent être trouvées dans les deux références précédentes ou dans <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> George Campbell ; Ronald Foster, <cite class="italique" lang="en">Fourier Integrals for Practical Applications</cite>, New York, USA, D. Van Nostrand Company, Inc, <time>1948</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Integrals+for+Practical+Applications&rft.place=New+York%2C+USA&rft.pub=D.+Van+Nostrand+Company%2C+Inc&rft.au=George+Campbell+%3B+Ronald+Foster&rft.date=1948&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">. <table class="wikitable"> <tbody><tr> <th>Fonction</th> <th>Transformée de Fourier ξ est la fréquence </th> <th>Transformée de Fourier ω = 2πξ est la pulsation ou fréquence angulaire</th> <th>Transformée de Fourier définition alternative </th> <th>Remarques </th></tr> <tr> <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"></mspace> </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 )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi ix\xi }\,\mathrm {d} 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> <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> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi ix\xi }\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c018b6f8ef03faaee00bf37ac895c9a76a284f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:22.653ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi ix\xi }\,\mathrm {d} x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} 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> <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> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b854b4e123327c9e2c853458df5a85ac54b2ad90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:26.727ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,\mathrm {d} x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} 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> <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>ν</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f174616e75f34407c919c10500bc9f117a72e69b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.936ex; margin-bottom: -0.236ex; width:20.922ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,\mathrm {d} x\end{aligned}}}" /> </td> <td> </td></tr> <tr> <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"></mspace> </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|}}\cdot \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> <mo>⋅</mo> <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|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4655bb6a84f51299144ee4aa4fdd5d646833f578" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.414ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \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}}}}\cdot \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> <mo>⋅</mo> <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}}}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9cca64c48afea293da151e9087ab3f84ecd01fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.454ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \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|}}\cdot \operatorname {sinc} \left({\frac {\nu }{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> <mo>⋅</mo> <mi>sinc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</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|}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535011614a42dfc6546be45eb4a3a3dc9a964a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.263ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right)}" /> </td> <td>Pour la fonction rectangulaire voir <a href="/wiki/Fonction_porte" title="Fonction porte">fonction porte</a> ; la fonction <a href="/wiki/Sinus_cardinal" title="Sinus cardinal">sinus cardinal normalisé</a> est définie ici par sinc(x) = sin(πx)/πx </td></tr> <tr> <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"></mspace> </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|}}\cdot \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> <mo>⋅</mo> <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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfdea6503ebe44e2ee666762c8a3e82d7730ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.794ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \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}}}}\cdot \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> <mo>⋅</mo> <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}}}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d251db0f5151ddc7b50adc3706e4d9fd2d71261d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.447ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \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|}}\cdot \operatorname {rect} \left({\frac {\nu }{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> <mo>⋅</mo> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</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|}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cabebabf94a40550d76df5a74649b52741d55c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.256ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right)}" /> </td> <td>Relation duale de la précédente. La <a href="/wiki/Fonction_porte" title="Fonction porte">fonction porte</a> est un filtre passe-bas idéal, et la fonction <a href="/wiki/Sinus_cardinal" title="Sinus cardinal">sinus cardinal</a> est la réponse impulsionnelle non causale d'un tel filtre. La fonction sinc est la fonction <a href="/wiki/Sinus_cardinal" title="Sinus cardinal">sinus cardinal</a> normalisée : sinc(x) = sin(πx)/πx </td></tr> <tr> <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|}}\cdot \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> <mo>⋅</mo> <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|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8271127271dcff060cf227c6b07eeb308d851386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.989ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \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}}}}\cdot \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> <mo>⋅</mo> <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}}}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2190c69cd1c369c1de05ac502d6d132be4899135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.029ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \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|}}\cdot \operatorname {tri} \left({\frac {\nu }{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> <mo>⋅</mo> <mi>tri</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</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|}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e099919754c65dc8ccfd968b765bad626f5703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.838ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right)}" /> </td> <td>La fonction tri(x) est la <a href="/wiki/Fonction_triangulaire" title="Fonction triangulaire">fonction triangulaire</a>. </td></tr> <tr> <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|}}\cdot \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> <mo>⋅</mo> <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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c71b6749db521ee7873d74e29f3a8ea2f4890d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.855ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \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}}}}\cdot \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> <mo>⋅</mo> <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}}}}\cdot \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/be4f3cb4332b9381cac8f591b0fee9b7cbce357b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.508ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \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|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{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> <mo>⋅</mo> <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>ν</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|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58dd3f654cf5eb39fd7769112f0d7a0490d72c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.317ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right)}" /> </td> <td>Relation duale de la précédente </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-ax}u(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-ax}u(x)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eae1ae87bbb446154985a91a6cc4de98f585e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.209ex; height:3.009ex;" alt="{\displaystyle {\rm {e}}^{-ax}u(x)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a+2\pi {\rm {i}}\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> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+2\pi {\rm {i}}\xi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c14f388ac026674c8539cce4ded53ebfb6d7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.078ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{a+2\pi {\rm {i}}\xi }}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+{\rm {i}}\omega )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada3dcf472640c9015adcfd617a4a805d8942e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.239ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+{\rm {i}}\omega )}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a+{\rm {i}}\nu }}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+{\rm {i}}\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe7fd67f217c705ead7e18c1c9fe1c2a4e5a066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.785ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{a+{\rm {i}}\nu }}}" /> </td> <td>La fonction u(x) est la <a href="/wiki/Fonction_de_Heaviside" title="Fonction de Heaviside">fonction marche de Heaviside</a> et a > 0. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-\alpha x^{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{-\alpha x^{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64af35d93a5e2903bf200d3b74ddb0bc3c099bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.754ex; height:3.009ex;" alt="{\displaystyle {\rm {e}}^{-\alpha x^{2}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot {\rm {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> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 }}}\cdot {\rm {e}}^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ebfdae7d71e2600fcb7054458b3f7cbb8d7c66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.901ex; height:6.509ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot {\rm {e}}^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot {\rm {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> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 }}}\cdot {\rm {e}}^{-{\frac {\omega ^{2}}{4\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb5ae422f88f542e536d5c43d23fc9b1aa38f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.111ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot {\rm {e}}^{-{\frac {\omega ^{2}}{4\alpha }}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot {\rm {e}}^{-{\frac {\nu ^{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> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ν</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 }}}\cdot {\rm {e}}^{-{\frac {\nu ^{2}}{4\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b545bf02cfacd083c71c9088404f3d11938d153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.228ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot {\rm {e}}^{-{\frac {\nu ^{2}}{4\alpha }}}}" /> </td> <td>Nota : pour les deux premières transformations de Fourier, la <a href="/wiki/Fonction_gaussienne" title="Fonction gaussienne">fonction gaussienne</a> e−αx2 est, pour un choix judicieux de α, sa propre transformée de Fourier. Pour qu'elle soit intégrable on doit avoir Re(α) > 0. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {e} ^{-a|x|}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {e} ^{-a|x|}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91300f2c35cba357503651d5150640b6ff4a0f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.655ex; height:2.843ex;" alt="{\displaystyle \operatorname {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 }}}\cdot {\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> <mo>⋅</mo> <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 }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48073c2467507ea06912adf4d23576fd81c4e304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.632ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2a}{a^{2}+\nu ^{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>ν</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}+\nu ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afdbfc2d5d91d02ed543fae305c333d5e57bc8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.272ex; height:5.676ex;" alt="{\displaystyle {\frac {2a}{a^{2}+\nu ^{2}}}}" /> </td> <td>Pour Re(a) > 0. Ceci signifie que la transformée de Fourier d'une densité de probabilité d'une <a href="/wiki/Loi_de_Laplace_(probabilit%C3%A9s)" title="Loi de Laplace (probabilités)">distribution de Laplace</a> est une densité de probabilité d'une <a href="/wiki/Loi_de_Cauchy_(probabilit%C3%A9s)" title="Loi de Cauchy (probabilités)">loi de Cauchy</a>. </td></tr> <tr> <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"></mspace> </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}}\nu \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>ν</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7680a875d80fe4f647d8a1b16a4bb27235b824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.065ex; height:4.843ex;" alt="{\displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right)}" /> </td> <td>La <a href="/wiki/Fonction_hyperbolique#Sécante_hyperbolique" title="Fonction hyperbolique">sécante hyperbolique</a> est sa propre transformée de Fourier. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-{\frac {a^{2}x^{2}}{2}}}H_{n}(ax)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {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/f27808b99950737836d4ce08f0dceddf7e9972ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.356ex; height:4.509ex;" alt="{\displaystyle {\rm {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 }}(-{\rm {i}})^{n}}{a}}{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 }}(-{\rm {i}})^{n}}{a}}{\rm {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/db6efaa5625b10c339492aaef5d5af00248bc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.075ex; height:6.843ex;" alt="{\displaystyle {\frac {{\sqrt {2\pi }}(-{\rm {i}})^{n}}{a}}{\rm {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 {(-{\rm {i}})^{n}}{a}}{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 {(-{\rm {i}})^{n}}{a}}{\rm {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/1be0c7822963178e35ec53df6ee84a041c145d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.466ex; height:6.009ex;" alt="{\displaystyle {\frac {(-{\rm {i}})^{n}}{a}}{\rm {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 {(-{\rm {i}})^{n}{\sqrt {2\pi }}}{a}}{\rm {e}}^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ν</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>ν</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(-{\rm {i}})^{n}{\sqrt {2\pi }}}{a}}{\rm {e}}^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{a}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5a06a71b0ae35a34b04b65598cf07e1319adad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.683ex; height:6.009ex;" alt="{\displaystyle {\frac {(-{\rm {i}})^{n}{\sqrt {2\pi }}}{a}}{\rm {e}}^{-{\frac {\nu ^{2}}{2a^{2}}}}H_{n}\left({\frac {\nu }{a}}\right)}" /> </td> <td>Hn est le ne <a href="/wiki/Polyn%C3%B4me_d%27Hermite" title="Polynôme d'Hermite">polynôme d'Hermite</a>. Si a = 1 alors les <a href="/wiki/Polyn%C3%B4me_d%27Hermite#Fonctions_d'Hermite-Gauss" title="Polynôme d'Hermite">fonctions d'Hermite-Gauss</a> sont des <a href="/wiki/Fonction_propre" title="Fonction propre">fonctions propres</a> de l'opérateur transformée de Fourier. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Distributions_à_une_variable"><a href="/wiki/Distribution_(math%C3%A9matiques)" title="Distribution (mathématiques)">Distributions</a> à une variable</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=30" title="Modifier la section : Distributions à une variable" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=30" title="Modifier le code source de la section : Distributions à une variable">modifier le code</a>]</div> Les transformées de Fourier de ce tableau sont traitées dans <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Arthur Erdélyi, <cite class="italique" lang="en">Tables of Integral Transforms, Vol. 1</cite>, McGraw-Hill, <time>1954</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tables+of+Integral+Transforms%2C+Vol.+1&rft.pub=McGraw-Hill&rft.aulast=Erd%C3%A9lyi&rft.aufirst=Arthur&rft.date=1954&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier"> ou dans <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> David Kammler, <cite class="italique" lang="en">A First Course in Fourier Analysis</cite>, USA, Prentice Hall, <time>2000</time><span class="Z3988" 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.place=USA&rft.pub=Prentice+Hall&rft.aulast=Kammler&rft.aufirst=David&rft.date=2000&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">. <table class="wikitable"> <tbody><tr> <th>Fonction</th> <th>Transformée de Fourier ξ est la fréquence </th> <th>Transformée de Fourier ω = 2πξ est la pulsation ou fréquence angulaire</th> <th>Transformée de Fourier définition alternative </th> <th>Remarques </th></tr> <tr> <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"></mspace> </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 )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,{\rm {d}}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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>x</mi> <mi>ξ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,{\rm {d}}x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fd055e6f916f396546b527556ba20ccfd5a864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.159ex; margin-bottom: -0.179ex; width:27.082ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-2\pi {\rm {i}}x\xi }\,{\rm {d}}x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,{\rm {d}}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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,{\rm {d}}x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c164079599527b257bfd16911d9718bd02c67721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.681ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\omega x}\,{\rm {d}}x\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,{\rm {d}}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> <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>ν</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,{\rm {d}}x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d119630b07c290aeea7fce8d27f6e4b9f06cb0f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.159ex; margin-bottom: -0.179ex; width:25.663ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )=\int _{-\infty }^{\infty }f(x){\rm {e}}^{-{\rm {i}}\nu x}\,{\rm {d}}x\end{aligned}}}" /> </td> <td> </td></tr> <tr> <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 }}\cdot \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> <mo>⋅</mo> <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 }}\cdot \delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5f9f37c14b7cab11716f08fa9aff2259898bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.413ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \delta (\nu )}"> <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>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \delta (\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929771fd856c95a104c19963a8851646535d2542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.585ex; height:2.843ex;" alt="{\displaystyle 2\pi \delta (\nu )}" /> </td> <td>δ(ξ) est la <a href="/wiki/Distribution_de_Dirac" title="Distribution de Dirac">distribution de Dirac</a>. </td></tr> <tr> <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"></mspace> </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"></mspace> </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>Relation duale de la précédente. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{{\rm {i}}ax}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {e}}^{{\rm {i}}ax}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd13306ff952c19efffca909b1cfb6421969442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.532ex; height:2.676ex;" alt="{\displaystyle {\rm {e}}^{{\rm {i}}ax}}" /> </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 }}\cdot \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> <mo>⋅</mo> <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 }}\cdot \delta (\omega -a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a285410b0a5cb1a716b3a5ebdefba3abdaeca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.483ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \delta (\nu -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>ν</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \delta (\nu -a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789f376bc5c4909046fdde19510be4553cab4c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.655ex; height:2.843ex;" alt="{\displaystyle 2\pi \delta (\nu -a)}" /> </td></tr> <tr> <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 }}\cdot {\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> <mo>⋅</mo> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15f3f05db9d139ab083e3cd7908cd3dd46d742d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.921ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \left(\delta (\nu -a)+\delta (\nu +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>ν</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</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 (\nu -a)+\delta (\nu +a)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0298e8db931e60004f12c1f3b4fd9bd9f4816442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.689ex; height:2.843ex;" alt="{\displaystyle \pi \left(\delta (\nu -a)+\delta (\nu +a)\right)}" /> </td> <td>Résulte de la <a href="/wiki/Formule_d%27Euler" title="Formule d'Euler">formule d'Euler</a>. </td></tr> <tr> <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 }}\cdot {\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> <mo>⋅</mo> <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 }}\cdot {\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01cf05a52555d264a55be4a31dd216e907dd6b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.534ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}\pi \left(\delta (\nu -a)-\delta (\nu +a)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\rm {i}}\pi \left(\delta (\nu -a)-\delta (\nu +a)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb451a45024148ad1416c337a72963e856825b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.144ex; height:2.843ex;" alt="{\displaystyle -{\rm {i}}\pi \left(\delta (\nu -a)-\delta (\nu +a)\right)}" /> </td> <td>Résulte de la <a href="/wiki/Formule_d%27Euler" title="Formule d'Euler">formule d'Euler</a>. </td></tr> <tr> <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 {\nu ^{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>ν</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 {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f6e03cd062401136089a2876ca5d9506e0896b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.648ex; height:6.509ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}" /> </td></tr> <tr> <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> <mspace width="thinmathspace"></mspace> </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/505fa58726646616e8ad49a9edb4d5b7ca1fdf46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.986ex; 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 {\nu ^{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>ν</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 {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee70e7f497f191e2e6e635ae9c4402e6d06ad46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.201ex; height:6.509ex;" alt="{\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}" /> </td> <td> </td></tr> <tr> <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"></mspace> </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 {\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\rm {i}}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8c538bb8f21b1cdd9770983dc53c4d98f46be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.748ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\rm {i}}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {i}}^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {i}}^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2020c0e9faef970a5121b6cf46412516e236e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.489ex; height:3.343ex;" alt="{\displaystyle {\rm {i}}^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi {\rm {i}}^{n}\delta ^{(n)}(\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </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>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi {\rm {i}}^{n}\delta ^{(n)}(\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9469bdf80a138659e4c9e8f4bb820c2276a9e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.34ex; height:3.343ex;" alt="{\displaystyle 2\pi {\rm {i}}^{n}\delta ^{(n)}(\nu )\,}" /> </td> <td>Ici n est un <a href="/wiki/Entier_naturel" title="Entier naturel">entier naturel</a> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(n)}(\xi )\,}"> <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>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ^{(n)}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8063f728b7008c8aca2caceed95a7fdd336b716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.778ex; height:3.343ex;" alt="{\displaystyle \delta ^{(n)}(\xi )\,}" /> est la nième dérivée (au sens des distributions) de la distribution de Dirac. On peut en déduire les transformées de tous les <a href="/wiki/Polyn%C3%B4me" title="Polynôme">polynômes</a>. </td></tr> <tr> <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> <mspace width="thinmathspace"></mspace> </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/689b8b937592b37f0288213fc894314dbac0e8d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.078ex; height:3.343ex;" alt="{\displaystyle \delta ^{(n)}(x)\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2\pi {\rm {i}}\xi )^{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ξ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2\pi {\rm {i}}\xi )^{n}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c53a13c77de8f989e56c4b8c4ecf7cfbd8b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.586ex; height:2.843ex;" alt="{\displaystyle (2\pi {\rm {i}}\xi )^{n}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {({\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {({\rm {i}}\omega )^{n}}{\sqrt {2\pi }}}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3863ad232751c54ec20670a400a474b754289974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:6.344ex; height:6.676ex;" alt="{\displaystyle {\frac {({\rm {i}}\omega )^{n}}{\sqrt {2\pi }}}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\rm {i}}\nu )^{n}\,}"> <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 mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\rm {i}}\nu )^{n}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cf6ebf02a614a65028cee72d709c331c62ff8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.294ex; height:2.843ex;" alt="{\displaystyle ({\rm {i}}\nu )^{n}\,}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(n)}(\xi )\,}"> <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>ξ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ^{(n)}(\xi )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8063f728b7008c8aca2caceed95a7fdd336b716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.778ex; height:3.343ex;" alt="{\displaystyle \delta ^{(n)}(\xi )\,}" /> est la nième dérivée (au sens des distributions) de la distribution de Dirac. </td></tr> <tr> <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 -{\rm {i}}\pi \operatorname {sgn}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 -{\rm {i}}\pi \operatorname {sgn}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd639be2319a7b5719ff733b0b7b9266303d5068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle -{\rm {i}}\pi \operatorname {sgn}(\xi )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b35ddb811b6cfb18444a54bc26990ddeb76373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.96ex; height:6.343ex;" alt="{\displaystyle -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}\pi \operatorname {sgn}(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\rm {i}}\pi \operatorname {sgn}(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2399ce356352ceb91c250230fbf818048323138e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.587ex; height:2.843ex;" alt="{\displaystyle -{\rm {i}}\pi \operatorname {sgn}(\nu )}" /> </td> <td>Ici sgn(ξ) est la <a href="/wiki/Fonction_signe" title="Fonction signe">fonction signe</a>. On notera que 1/x n'est pas une distribution. On doit utiliser la <a href="/wiki/Valeur_principale_de_Cauchy" title="Valeur principale de Cauchy">valeur principale de Cauchy</a> pour étudier les <a href="/wiki/Espace_de_Schwartz" title="Espace de Schwartz">fonctions de Schwartz</a>. Cette règle est utile quand on étudie la <a href="/wiki/Transformation_de_Hilbert" title="Transformation de Hilbert">transformation de Hilbert</a>. </td></tr> <tr> <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 {{\rm {d}}^{n}}{{\rm {d}}x^{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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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 {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\log |x|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b0541143cc755ea053f6acaf9c96e4ac6ae29b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:24.478ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}&{\frac {1}{x^{n}}}\\&:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\log |x|\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}\pi {\frac {(-2\pi {\rm {i}}\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 -{\rm {i}}\pi {\frac {(-2\pi {\rm {i}}\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236f4799de7254979bee61c6be9c2dae8f8303c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.329ex; height:6.676ex;" alt="{\displaystyle -{\rm {i}}\pi {\frac {(-2\pi {\rm {i}}\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-{\rm {i}}\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-{\rm {i}}\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497aa969b0d967eedd8d9db6ed84e32c710043cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.505ex; height:6.676ex;" alt="{\displaystyle -{\rm {i}}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-{\rm {i}}\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\rm {i}}\pi {\frac {(-{\rm {i}}\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</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>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\rm {i}}\pi {\frac {(-{\rm {i}}\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6312f53ca7031dea69a7563511d2fed38a386193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.238ex; height:6.676ex;" alt="{\displaystyle -{\rm {i}}\pi {\frac {(-{\rm {i}}\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )}" /> </td> <td>1/xn est la <a href="/w/index.php?title=Distribution_homog%C3%A8ne&action=edit&redlink=1" class="new" title="Distribution homogène (page inexistante)">distribution homogène</a> <a href="https://en.wikipedia.org/wiki/Homogeneous_distribution" class="extiw" title="en:Homogeneous distribution">(en)</a> définie par la dérivée de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {{\rm {d}}^{n}}{{\rm {d}}x^{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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <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 {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\log |x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ff02a898c09542567e446085b90e969dc30434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.981ex; height:6.676ex;" alt="{\displaystyle {\frac {(-1)^{n-1}}{(n-1)!}}{\frac {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\log |x|}" /> </td></tr> <tr> <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> <mspace width="thinmathspace"></mspace> </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/391c88fb6cfb5306b31728863a1da75f6af3ea9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.295ex; 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 }}}\cdot {\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> <mo>⋅</mo> <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 }}}\cdot {\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/ae775278ec202382905671cf8309f68da8f9b496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.35ex; height:7.343ex;" alt="{\displaystyle {\frac {-2}{\sqrt {2\pi }}}\cdot {\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)}{|\nu |^{\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>ν</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)}{|\nu |^{\alpha +1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c04a7346a0d14e7ee3a368e72f5702a8023a2b4" 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)}{|\nu |^{\alpha +1}}}}" /> </td> <td>Formule valide pour α réel avec -1 < α < 0. Si α complexe avec Re (α) > −1, alors <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> <mspace width="thinmathspace"></mspace> </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/391c88fb6cfb5306b31728863a1da75f6af3ea9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.295ex; height:3.009ex;" alt="{\displaystyle |x|^{\alpha }\,}" /> est une fonction localement intégrable et est donc une <a href="/wiki/Distribution_temp%C3%A9r%C3%A9e" title="Distribution tempérée">distribution tempérée</a>. La fonction α ↦ |x|α est une <a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">fonction holomorphe</a> du demi-plan complexe réel dans l'espace des distributions tempérées. Elle admet une unique extension <a href="/wiki/Fonction_m%C3%A9romorphe" title="Fonction méromorphe">méromorphe</a> qui est une distribution tempérée également notée <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> <mspace width="thinmathspace"></mspace> </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/391c88fb6cfb5306b31728863a1da75f6af3ea9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.295ex; height:3.009ex;" alt="{\displaystyle |x|^{\alpha }\,}" /> uniquement pour α ≠ −2, −4,.... </td></tr> <tr> <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> <mspace width="thinmathspace"></mspace> </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/8e6bd55de51b1769bb00522879d539a2150551ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.17ex; 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 {|\nu |}}}}"> <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>ν</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 {|\nu |}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effc1fd470644001a984cd045f7afc7ede8b2ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.686ex; height:7.176ex;" alt="{\displaystyle {\frac {\sqrt {2\pi }}{\sqrt {|\nu |}}}}" /> </td> <td>Cas particulier de la précédente, pour α = –1/2. </td></tr> <tr> <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}{{\rm {i}}\pi \xi }}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>ξ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\rm {i}}\pi \xi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d569450a9ec14f35b23e39816c95321edb02e442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.845ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{{\rm {i}}\pi \xi }}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ω</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{{\rm {i}}\omega }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952c39cd5f16be391a78089d3bc98b3cf97cc265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.421ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {1}{{\rm {i}}\omega }}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{{\rm {i}}\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>ν</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{{\rm {i}}\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a2a3710c17b82ecd7108e65880f0d5a5668709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.715ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{{\rm {i}}\nu }}}" /> </td> <td>La transformation de Fourier doit ici être prise comme la <a href="/wiki/Valeur_principale_de_Cauchy" title="Valeur principale de Cauchy">valeur principale de Cauchy</a>. </td></tr> <tr> <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}{{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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}{{\rm {i}}\pi \xi }}+\delta (\xi )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b280220398c6cdc7e0cc4bcd7732ce61e75cb201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.38ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2}}\left({\frac {1}{{\rm {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}{{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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}{{\rm {i}}\pi \omega }}+\delta (\omega )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10fbc05d4153f2f788d68e680b18330d568e60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.705ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{{\rm {i}}\pi \omega }}+\delta (\omega )\right)}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \left({\frac {1}{{\rm {i}}\pi \nu }}+\delta (\nu )\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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>π</mi> <mi>ν</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \left({\frac {1}{{\rm {i}}\pi \nu }}+\delta (\nu )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2dacdaaf13031089ae0cef023ce3b819ca4bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.118ex; height:6.176ex;" alt="{\displaystyle \pi \left({\frac {1}{{\rm {i}}\pi \nu }}+\delta (\nu )\right)}" /> </td> <td>La fonction u(x) est la <a href="/wiki/Fonction_de_Heaviside" title="Fonction de Heaviside">fonction de Heaviside</a>. </td></tr> <tr> <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(\nu -{\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>ν</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(\nu -{\frac {2\pi k}{T}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86d0d9e88fe02df56d8d713464611b88c3c3ad18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.633ex; height:7.009ex;" alt="{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {2\pi k}{T}}\right)}" /> </td> <td>Transformée de Fourier du <a href="/wiki/Peigne_de_Dirac" title="Peigne de Dirac">peigne de Dirac</a>. On utilise aussi le fait que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }{\rm {e}}^{{\rm {i}}nx}=2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>n</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }{\rm {e}}^{{\rm {i}}nx}=2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f815232819723f3ea013c5a78184f61c7b83942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.38ex; height:7.009ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }{\rm {e}}^{{\rm {i}}nx}=2\pi \sum _{k=-\infty }^{\infty }\delta (x+2\pi k)}" /> sont considérées comme des distributions. </td></tr> <tr> <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"></mspace> <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 }}}\cdot {\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> <mo>⋅</mo> <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 }}}\cdot {\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/58c4167455c507511e62c91249394dcfb5afff35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.834ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\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 {\nu }{2}}\right)}{\sqrt {1-\nu ^{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"></mspace> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ν</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 {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/839964293a0512ec275053b4d6dd9b5e99788c90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.491ex; height:7.509ex;" alt="{\displaystyle {\frac {2\,\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}" /> </td> <td>La fonction J0(x) est la <a href="/wiki/Fonction_de_Bessel" title="Fonction de Bessel">fonction de Bessel</a> d'ordre zéro de la <abbr class="abbr" title="Première">1re</abbr> espèce. </td></tr> <tr> <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(-{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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(-{\rm {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/8c92dc157d7e5c365aa8f02122cc70fc4faaa94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.831ex; height:7.009ex;" alt="{\displaystyle {\frac {2(-{\rm {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 {(-{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 {(-{\rm {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/53c4b386d2928e9cb2e6ec6b3990a7309c68a5c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.898ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-{\rm {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(-{\rm {i}})^{n}T_{n}(\nu )\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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>ν</mi> <mo stretchy="false">)</mo> <mi>rect</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2(-{\rm {i}})^{n}T_{n}(\nu )\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8693f6400774d0aef67b537dffa6d6e1ea023b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.204ex; height:7.509ex;" alt="{\displaystyle {\frac {2(-{\rm {i}})^{n}T_{n}(\nu )\operatorname {rect} \left({\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}" /> </td> <td>Généralisation de 315. La fonction Jn(x) est la <a href="/wiki/Fonction_de_Bessel" title="Fonction de Bessel">fonction de Bessel</a> d'ordre n de la <abbr class="abbr" title="Première">1re</abbr> espèce. La fonction Tn(x) est le <a href="/wiki/Polyn%C3%B4me_de_Tchebychev" title="Polynôme de Tchebychev">polynôme de Tchebychev</a> de <abbr class="abbr" title="Première">1re</abbr> espèce. </td></tr> <tr> <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 (\xi )}"> <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> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta (\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7580363a41198135e7ac5de920ae465f225b914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.957ex; height:6.009ex;" alt="{\displaystyle -{\frac {1}{2}}{\frac {1}{\left|\xi \right|}}-\gamma \delta (\xi )}" /> </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 (\omega )}"> <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> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta (\omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e281740c4693ea77d8079425e5d5191c57313ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.583ex; height:8.509ex;" alt="{\displaystyle -{\frac {\sqrt {\frac {\pi }{2}}}{\left|\omega \right|}}-{\sqrt {2\pi }}\gamma \delta (\omega )}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{\left|\nu \right|}}-2\pi \gamma \delta (\nu )}"> <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>ν</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>γ</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{\left|\nu \right|}}-2\pi \gamma \delta (\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a12c01616b5a7319f07af8cf5161742c521c5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.857ex; height:5.509ex;" alt="{\displaystyle -{\frac {\pi }{\left|\nu \right|}}-2\pi \gamma \delta (\nu )}" /> </td> <td>γ est la <a href="/wiki/Constante_d%27Euler-Mascheroni" title="Constante d'Euler-Mascheroni">constante d'Euler-Mascheroni</a>. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mp {\rm {i}}x\right)^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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 {\rm {i}}x\right)^{-\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c31571f8a3b121e57c23b352aab0e41baa4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.157ex; height:3.176ex;" alt="{\displaystyle \left(\mp {\rm {i}}x\right)^{-\alpha }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(2\pi \right)^{\alpha }}{\Gamma (\alpha )}}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> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </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 (\alpha )}}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/773d851e75202854d1202c3e1abb4e0467c461c4" 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 (\alpha )}}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 (\alpha )}}u(\pm \omega )(\pm \omega )^{\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> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mo>±</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <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 (\alpha )}}u(\pm \omega )(\pm \omega )^{\alpha -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae432a2bc590c18a5ac4f7878c8e37cda761c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.427ex; height:6.676ex;" alt="{\displaystyle {\frac {\sqrt {2\pi }}{\Gamma (\alpha )}}u(\pm \omega )(\pm \omega )^{\alpha -1}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u(\pm \nu )(\pm \nu )^{\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> <mo stretchy="false">(</mo> <mo>±</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mi>ν</mi> <msup> <mo stretchy="false">)</mo> <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(\pm \nu )(\pm \nu )^{\alpha -1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ab036b85be810bfd6977ce422c96a58d8e5a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.386ex; height:6.009ex;" alt="{\displaystyle {\frac {2\pi }{\Gamma \left(\alpha \right)}}u(\pm \nu )(\pm \nu )^{\alpha -1}}" /> </td> <td>Formule correcte pour 0 < α < 1. La formule de dérivation permet de déduire la formule pour des exposants plus élevés. u est la fonction de Heaviside. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Fonctions_de_deux_variables">Fonctions de deux variables</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=31" title="Modifier la section : Fonctions de deux variables" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=31" title="Modifier le code source de la section : Fonctions de deux variables">modifier le code</a>]</div> <table class="wikitable"> <tbody><tr> <th>Fonction</th> <th>Transformée de Fourier ξ est la fréquence </th> <th>Transformée de Fourier ω = 2πξ est la pulsation ou fréquence angulaire</th> <th>Transformée de Fourier définition alternative </th> <th>Remarques </th></tr> <tr> <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})\\&=\iint f(x,y){\rm {e}}^{-2\pi {\rm {i}}(\xi _{x}x+\xi _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\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> <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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\\&=\iint f(x,y){\rm {e}}^{-2\pi {\rm {i}}(\xi _{x}x+\xi _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770226f7b84b9bc06a66f1ffb0e7d0c6e511fcf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.384ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\xi _{x},\xi _{y})\\&=\iint f(x,y){\rm {e}}^{-2\pi {\rm {i}}(\xi _{x}x+\xi _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\\&={\frac {1}{2\pi }}\iint f(x,y){\rm {e}}^{-{\rm {i}}(\omega _{x}x+\omega _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\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> <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> </mtd> </mtr> <mtr> <mtd></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> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\\&={\frac {1}{2\pi }}\iint f(x,y){\rm {e}}^{-{\rm {i}}(\omega _{x}x+\omega _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd978124efb8c4a1c5aff32fffc7bb86a4bee5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:34.942ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\omega _{x},\omega _{y})\\&={\frac {1}{2\pi }}\iint f(x,y){\rm {e}}^{-{\rm {i}}(\omega _{x}x+\omega _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu _{x},\nu _{y})\\&=\iint f(x,y){\rm {e}}^{-{\rm {i}}(\nu _{x}x+\nu _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\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> <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>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∬</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\hat {f}}(\nu _{x},\nu _{y})\\&=\iint f(x,y){\rm {e}}^{-{\rm {i}}(\nu _{x}x+\nu _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35765cd682acc3828a76cd6ea85ebcb1f0be7d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.804ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&{\hat {f}}(\nu _{x},\nu _{y})\\&=\iint f(x,y){\rm {e}}^{-{\rm {i}}(\nu _{x}x+\nu _{y}y)}\,{\rm {d}}x\,{\rm {d}}y\end{aligned}}}" /> </td> <td>Les variables ξx, ξy, ωx, ωy, νx, νy sont réelles. Les intégrales couvrent tout le plan complexe. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {e}}^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 {\rm {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/e5717a59798efc53c764825c4eda342276eff9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.932ex; height:3.009ex;" alt="{\displaystyle {\rm {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|}}{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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|}}{\rm {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/4871ddafacf32ff1c985c0ef926cf48c2584bb84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.297ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{|ab|}}{\rm {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 \cdot |ab|}}{\rm {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> <mo>⋅</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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 \cdot |ab|}}{\rm {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/7ef73e091c9e8334e4d213e541492418f952423e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.253ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{2\pi \cdot |ab|}}{\rm {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|}}{\rm {e}}^{-{\frac {1}{4\pi }}\left({\frac {\nu _{x}^{2}}{a^{2}}}+{\frac {\nu _{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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <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>ν</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>ν</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|}}{\rm {e}}^{-{\frac {1}{4\pi }}\left({\frac {\nu _{x}^{2}}{a^{2}}}+{\frac {\nu _{y}^{2}}{b^{2}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb11bb69f52735f99dc66634a67d98360a25873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.801ex; height:8.176ex;" alt="{\displaystyle {\frac {1}{|ab|}}{\rm {e}}^{-{\frac {1}{4\pi }}\left({\frac {\nu _{x}^{2}}{a^{2}}}+{\frac {\nu _{y}^{2}}{b^{2}}}\right)}}" /> </td> <td>La fonction et ses transformées sont toutes des gaussiennes. </td></tr> <tr> <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 {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{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>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ν</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>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ν</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 {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bca6577770939d05b3910eccaa333adbfc78fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:18.634ex; height:10.509ex;" alt="{\displaystyle {\frac {2\pi J_{1}\left({\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}}" /> </td> <td>La fonction est définie par circ(r) = 1 sur 0 ≤ r ≤ 1, et est nulle partout ailleurs. Le résultat est la distribution de l'amplitude de la <a href="/wiki/Tache_d%27Airy" title="Tache d'Airy">tache d'Airy</a>. J1 est la <a href="/wiki/Fonction_de_Bessel" title="Fonction de Bessel">fonction de Bessel</a> de première espèce d'ordre 1<a href="#cite_note-6">[6]</a>. </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\rm {i}}{x+{\rm {i}}y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {i}}{x+{\rm {i}}y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2168fa98b4c1c8003c1d1903d01dbe8409b1fe43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.809ex; height:5.676ex;" alt="{\displaystyle {\frac {\rm {i}}{x+{\rm {i}}y}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\xi _{x}+{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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}+{\rm {i}}\xi _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d44ccee42ad793aace6936fa7602fd83d1a684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.582ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\xi _{x}+{\rm {i}}\xi _{y}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\omega _{x}+{\rm {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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <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}+{\rm {i}}\omega _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9558ba37718ce9e4e292d95cb8785c772ad40f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.437ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{\omega _{x}+{\rm {i}}\omega _{y}}}}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{\nu _{x}+{\rm {i}}\nu _{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>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msub> <mi>ν</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 }{\nu _{x}+{\rm {i}}\nu _{y}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096ce8aff0034ad31d8c603055907bf4654e6d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.842ex; height:5.843ex;" alt="{\displaystyle {\frac {2\pi }{\nu _{x}+{\rm {i}}\nu _{y}}}}" /> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références">Notes et références</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=32" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=32" title="Modifier le code source de la section : Notes et références">modifier le code</a>]</div> <div style="font-size:85%; padding-left:1.6em; margin:0.3em 0;"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « <a class="external text" href="https://en.wikipedia.org/wiki/Fourier_transform?oldid=583088122">Fourier transform</a> » (<a class="external text" href="https://en.wikipedia.org/wiki/Fourier_transform?action=history">voir la liste des auteurs</a>).</div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="/wiki/Commission_%C3%A9lectrotechnique_internationale" title="Commission électrotechnique internationale">Commission électrotechnique internationale</a>, <cite style="font-style:normal">« Mathématiques - Fonctions : Transformations integrales »</cite>, dans <cite class="italique">IEC 60050 Vocabulaire électrotechnique international</cite>, 1987/1994 (<a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=103-04-01">lire en ligne</a>), <abbr class="abbr" title="page(s)">p.</abbr> 103-04-01<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=IEC+60050+Vocabulaire+%C3%A9lectrotechnique+international&rft.atitle=Math%C3%A9matiques+-+Fonctions+%3A+Transformations++integrales&rft.aucorp=%5B%5BCommission+%C3%A9lectrotechnique+internationale%5D%5D&rft.date=1994&rft.pages=103-04-01&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier"> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="/wiki/Alain_Bouvier" title="Alain Bouvier">Alain Bouvier</a>, Michel George et <a href="/wiki/Fran%C3%A7ois_Le_Lionnais" title="François Le Lionnais">François Le Lionnais</a>, <cite class="italique">Dictionnaire des mathématiques</cite>, <a href="/wiki/Presses_universitaires_de_France" title="Presses universitaires de France">Presses universitaires de France</a>, <time>2001</time> (<abbr class="abbr" title="première">1re</abbr> <abbr class="abbr" title="édition">éd.</abbr> 1979), <abbr class="abbr" title="page">p.</abbr> 361<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionnaire+des+math%C3%A9matiques&rft.pub=Presses+universitaires+de+France&rft.aulast=Bouvier&rft.aufirst=Alain&rft.au=George%2C+Michel&rft.au=Le+Lionnais%2C+Fran%C3%A7ois&rft.date=2001&rft.pages=361&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">. </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a href="/wiki/Walter_Rudin" title="Walter Rudin">Walter Rudin</a>, <cite class="italique">Analyse réelle et complexe</cite> [<a href="/wiki/R%C3%A9f%C3%A9rence:Analyse_(Rudin)" title="Référence:Analyse (Rudin)">détail des éditions</a>]<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analyse+r%C3%A9elle+et+complexe&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">, p. 174 de l'édition de 1975-77. </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/w/index.php?title=Mark_Pinsky&action=edit&redlink=1" class="new" title="Mark Pinsky (page inexistante)">Mark Pinsky</a> <a href="https://en.wikipedia.org/wiki/Mark_Pinsky" class="extiw" title="en:Mark Pinsky">(en)</a>, <cite class="italique" lang="en">Introduction to Fourier Analysis and Wavelets</cite>, Brooks/Cole, <time>2002</time> (<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-8218-7198-0" title="Spécial:Ouvrages de référence/978-0-8218-7198-0">978-0-8218-7198-0</a>, <a rel="nofollow" class="external text" href="//books.google.com/books?id=tlLE4KUkk1gC&pg=PA131">lire en ligne</a>), <abbr class="abbr" title="page">p.</abbr> 131<span class="Z3988" 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.aulast=%3AMark+Pinsky&rft.date=2002&rft.pages=131&rft.isbn=978-0-8218-7198-0&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATransformation+de+Fourier">. </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Ou plus précisément à l’ombre de cette somme<a href="/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" title="Aide:Référence nécessaire">[réf. nécessaire]</a>. </li> <li id="cite_note-6"><a href="#cite_ref-6">↑</a> <a href="#SteinWeiss1971">Stein et Weiss 1971</a>, Thm. IV.3.3. </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=33" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=33" title="Modifier le code source de la section : Voir aussi">modifier le code</a>]</div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> Sur les autres projets Wikimedia : <ul class="noarchive plainlinks"> <li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/S%C3%A9rie_et_transform%C3%A9e_de_Fourier_en_physique" class="extiw" title="v:Série et transformée de Fourier en physique">Série et transformée de Fourier en physique</a>, sur Wikiversity</li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=34" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=34" title="Modifier le code source de la section : Articles connexes">modifier le code</a>]</div> <ul><li><a href="/wiki/Densit%C3%A9_spectrale" title="Densité spectrale">Densité spectrale</a></li> <li><a href="/wiki/Densit%C3%A9_spectrale_de_puissance" title="Densité spectrale de puissance">Densité spectrale de puissance</a></li> <li><a href="/wiki/Produit_de_convolution" title="Produit de convolution">Produit de convolution</a></li> <li><a href="/wiki/Transformation_de_Fourier_rapide" title="Transformation de Fourier rapide">Transformation de Fourier rapide</a></li> <li><a href="/wiki/Transformation_de_Fourier_discr%C3%A8te" title="Transformation de Fourier discrète">Transformation de Fourier discrète</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_d%27inversion_de_Fourier" title="Théorème d'inversion de Fourier">Théorème d'inversion de Fourier</a></li> <li><a href="/wiki/Transformation_de_Laplace" title="Transformation de Laplace">Transformation de Laplace</a></li> <li><a href="/wiki/Transformation_de_Hankel" title="Transformation de Hankel">Transformation de Hankel</a></li> <li><a href="/wiki/Transformation_de_Mellin" title="Transformation de Mellin">Transformation de Mellin</a></li> <li><a href="/wiki/Bispectre" title="Bispectre">Bispectre</a></li> <li><a href="/wiki/Spectroscopie_transform%C3%A9e_de_Fourier" class="mw-redirect" title="Spectroscopie transformée de Fourier">Spectroscopie transformée de Fourier</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=35" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=35" title="Modifier le code source de la section : Bibliographie">modifier le code</a>]</div> <ul><li><a href="/wiki/Jean-Michel_Bony" title="Jean-Michel Bony">Jean-Michel Bony</a>, Cours d'analyse, <a href="/wiki/%C3%89cole_polytechnique_(France)#Activités_de_recherche" title="École polytechnique (France)">Éditions de l'École Polytechnique</a></li> <li>Srishti D. Chatterji, Cours d'analyse, <a href="/wiki/Presses_polytechniques_et_universitaires_romandes" title="Presses polytechniques et universitaires romandes">Presses polytechniques et universitaires romandes</a>, 1998 (<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2880743468" title="Spécial:Ouvrages de référence/978-2880743468">978-2880743468</a>)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3>[<a href="/w/index.php?title=Transformation_de_Fourier&veaction=edit&section=36" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor">modifier</a> | <a href="/w/index.php?title=Transformation_de_Fourier&action=edit&section=36" title="Modifier le code source de la section : Liens externes">modifier le code</a>]</div> <ul><li>Alain Yger, <a rel="nofollow" class="external text" href="http://www.math.u-bordeaux1.fr/~yger/mht613.pdf">Espaces de Hilbert et analyse de Fourier</a> (2008), cours de <abbr class="abbr" title="Troisième">3e</abbr> année de licence, <a href="/wiki/Universit%C3%A9_Bordeaux_I" class="mw-redirect" title="Université Bordeaux I">université Bordeaux I</a></li> <li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> « <a rel="nofollow" class="external text" href="http://www.jcrystal.com/products/ftlse/index.htm"><cite style="font-style:normal;" lang="en">FTL-SE</cite></a> », programme éducatif sur les transformées de Fourier d'images</li> <li>J. 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III (fondements de la transformée de Fourier), en ligne et commenté sur le site <a href="/wiki/BibNum" class="mw-redirect" title="BibNum">BibNum</a></li> <li><a rel="nofollow" class="external text" href="http://lodel.irevues.inist.fr/oeiletphysiologiedelavision/index.php?id=203#f17">Œil et physiologie de la vision : « les signaux électrophysiologiques »</a>.</li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Analyse_fonctionnelle" title="Modèle:Palette Analyse fonctionnelle"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Analyse_fonctionnelle&action=edit"><abbr class="abbr" title="Modifier ce modèle. 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Transformation de Fourier — Wikipédia