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class="vector-toc-link" href="#Fourier_Analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Fourier Analysis</span> </div> </a> <ul id="toc-Fourier_Analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_harmonic_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Abstract_harmonic_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Abstract harmonic analysis</span> </div> </a> <ul id="toc-Abstract_harmonic_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applied_harmonic_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applied_harmonic_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applied harmonic analysis</span> </div> </a> <ul id="toc-Applied_harmonic_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_branches" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_branches"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other branches</span> </div> </a> <ul id="toc-Other_branches-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Major_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Major_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Major results</span> </div> </a> <ul id="toc-Major_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> 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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"><span class="mw-page-title-main">Harmonic analysis</span></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="Go to an article in another language. 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mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_harm%C3%B3nicu" title="Analís harmónicu – Asturian" lang="ast" hreflang="ast" data-title="Analís harmónicu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Harmonik_analiz" title="Harmonik analiz – Azerbaijani" lang="az" hreflang="az" data-title="Harmonik analiz" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BA_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Гармоник анализ – Bashkir" lang="ba" hreflang="ba" data-title="Гармоник анализ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Гарманічны аналіз – Belarusian" lang="be" hreflang="be" data-title="Гарманічны аналіз" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Хармоничен анализ – Bulgarian" lang="bg" hreflang="bg" data-title="Хармоничен анализ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_harm%C3%B2nica" title="Anàlisi harmònica – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi harmònica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Гармонилле анализ – Chuvash" lang="cv" hreflang="cv" data-title="Гармонилле анализ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Harmonick%C3%A1_anal%C3%BDza" title="Harmonická analýza – Czech" lang="cs" hreflang="cs" data-title="Harmonická analýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Harmonische_Analyse" title="Harmonische Analyse – German" lang="de" hreflang="de" data-title="Harmonische Analyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Harmooniline_anal%C3%BC%C3%BCs" title="Harmooniline analüüs – Estonian" lang="et" hreflang="et" data-title="Harmooniline analüüs" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_arm%C3%B3nico" title="Análisis armónico – Spanish" lang="es" hreflang="es" data-title="Análisis armónico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo badge-Q70894304 mw-list-item" title=""><a href="https://eo.wikipedia.org/wiki/Harmona_analitiko" title="Harmona analitiko – Esperanto" lang="eo" hreflang="eo" data-title="Harmona analitiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Analisi_harmoniko" title="Analisi harmoniko – Basque" lang="eu" hreflang="eu" data-title="Analisi harmoniko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D9%87%D8%A7%D8%B1%D9%85%D9%88%D9%86%DB%8C%DA%A9" title="آنالیز هارمونیک – Persian" lang="fa" hreflang="fa" data-title="آنالیز هارمونیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_harmonique_(math%C3%A9matiques)" title="Analyse harmonique (mathématiques) – French" lang="fr" hreflang="fr" data-title="Analyse harmonique (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_harm%C3%B3nica" title="Análise harmónica – Galician" lang="gl" hreflang="gl" data-title="Análise harmónica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A1%B0%ED%99%94%ED%95%B4%EC%84%9D%ED%95%99" title="조화해석학 – Korean" lang="ko" hreflang="ko" data-title="조화해석학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D6%80%D5%B4%D5%B8%D5%B6%D5%AB%D5%AF_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Հարմոնիկ անալիզ – Armenian" lang="hy" hreflang="hy" data-title="Հարմոնիկ անալիզ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AE%E0%A5%8B%E0%A4%A8%E0%A4%BF%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="हार्मोनिक विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="हार्मोनिक विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_armonica" title="Analisi armonica – Italian" lang="it" hreflang="it" data-title="Analisi armonica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%94%D7%A8%D7%9E%D7%95%D7%A0%D7%99%D7%AA" title="אנליזה הרמונית – Hebrew" lang="he" hreflang="he" data-title="אנליזה הרמונית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Harmonikus_anal%C3%ADzis" title="Harmonikus analízis – Hungarian" lang="hu" hreflang="hu" data-title="Harmonikus analízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Analisi_armonika" title="Analisi armonika – Maltese" lang="mt" hreflang="mt" data-title="Analisi armonika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Harmonische_analyse" title="Harmonische analyse – Dutch" lang="nl" hreflang="nl" data-title="Harmonische analyse" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AA%BF%E5%92%8C%E8%A7%A3%E6%9E%90" title="調和解析 – Japanese" lang="ja" hreflang="ja" data-title="調和解析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Harmonisk_analyse" title="Harmonisk analyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Harmonisk analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Garmonik_analiz" title="Garmonik analiz – Uzbek" lang="uz" hreflang="uz" data-title="Garmonik analiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_harmoniczna" title="Analiza harmoniczna – Polish" lang="pl" hreflang="pl" data-title="Analiza harmoniczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_harm%C3%B3nica" title="Análise harmónica – Portuguese" lang="pt" hreflang="pt" data-title="Análise harmónica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Analiz%C4%83_armonic%C4%83" title="Analiză armonică – Romanian" lang="ro" hreflang="ro" data-title="Analiză armonică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Гармонический анализ – Russian" lang="ru" hreflang="ru" data-title="Гармонический анализ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Analiza_harmonike" title="Analiza harmonike – Albanian" lang="sq" hreflang="sq" data-title="Analiza harmonike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Harmonic_analysis" title="Harmonic analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Harmonic analysis" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Harmonijska_analiza" title="Harmonijska analiza – Serbian" lang="sr" hreflang="sr" data-title="Harmonijska analiza" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Harmonik_analiz" title="Harmonik analiz – Turkish" lang="tr" hreflang="tr" data-title="Harmonik analiz" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Гармонічний аналіз – Ukrainian" lang="uk" hreflang="uk" data-title="Гармонічний аналіз" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%B0%83%E5%92%8C%E5%88%86%E6%9E%90" title="调和分析 – Wu" lang="wuu" hreflang="wuu" data-title="调和分析" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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searchaux" style="display:none">Study of superpositions in mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the process of determining the structure of a piece of music, see <a href="/wiki/Harmony" title="Harmony">Harmony</a>.</div> <p class="mw-empty-elt"> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Harmonic_(mathematics)" title="Harmonic (mathematics)">Harmonic (mathematics)</a>.</div> <p><b>Harmonic analysis</b> is a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> concerned with investigating the connections between a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> and its representation in <a href="/wiki/Frequency" title="Frequency">frequency</a>. The frequency representation is found by using the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> for functions on unbounded domains such as the full <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> or by <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> for functions on bounded domains, especially periodic functions on finite <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>. Generalizing these transforms to other domains is generally called <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Tidal_analysis" class="mw-redirect" title="Tidal analysis">tidal analysis</a>, <a href="/wiki/Spectral_analysis" class="mw-disambig" title="Spectral analysis">Spectral Analysis</a>, and <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>. </p><p>The term "<a href="/wiki/Harmonic" title="Harmonic">harmonics</a>" originated from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> word <i>harmonikos</i>, meaning "skilled in music".<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In physical <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> problems, it began to mean waves whose frequencies are <a href="/wiki/Multiple_(mathematics)" title="Multiple (mathematics)">integer multiples</a> of one another, as are the frequencies of the <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonics of music notes</a>. Still, the term has been generalized beyond its original meaning. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Development_of_Harmonic_Analysis">Development of Harmonic Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=1" title="Edit section: Development of Harmonic Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Historically, <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic functions</a> first referred to the solutions of <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This terminology was extended to other <a href="/wiki/Special_functions" title="Special functions">special functions</a> that solved related equations,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> then to <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of general <a href="/wiki/Elliptic_operator" title="Elliptic operator">elliptic operators</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and nowadays harmonic functions are considered as a generalization of periodic functions<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Function_space" title="Function space">function spaces</a> defined on <a href="/wiki/Manifold" title="Manifold">manifolds</a>, for example as solutions of general, not necessarily <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic</a>, <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> including some <a href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a> that may imply their symmetry or periodicity.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Fourier_Analysis">Fourier Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=2" title="Edit section: Fourier Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fourier_Analysis" class="mw-redirect" title="Fourier Analysis">Fourier Analysis</a></div> <p>The classical <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> on <b><a href="/wiki/Real_number" title="Real number">R</a></b><sup><i>n</i></sup> is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as <a href="/wiki/Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">tempered distributions</a>. For instance, if we impose some requirements on a distribution <i>f</i>, we can attempt to translate these requirements into the Fourier transform of <i>f</i>. The <a href="/wiki/Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorem</a> is an example. The Paley–Wiener theorem immediately implies that if <i>f</i> is a nonzero <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> of <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a> (these include functions of compact support), then its Fourier transform is never compactly supported (i.e., if a signal is limited in one domain, it is unlimited in the other). This is an elementary form of an <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> in a harmonic-analysis setting. </p><p>Fourier series can be conveniently studied in the context of <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, which provides a connection between harmonic analysis and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation: </p> <ul><li>Discrete/periodic–discrete/periodic: <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></li> <li>Continuous/periodic–discrete/aperiodic: <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li>Discrete/aperiodic–continuous/periodic: <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></li> <li>Continuous/aperiodic–continuous/aperiodic: <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li></ul> <p>As the spaces mapped by the Fourier transform are, in particular, subspaces of the space of tempered distributions it can be shown that the four versions of the Fourier transform are particular cases of the Fourier transform on tempered distributions. </p> <div class="mw-heading mw-heading2"><h2 id="Abstract_harmonic_analysis">Abstract harmonic analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=3" title="Edit section: Abstract harmonic analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Abstract harmonic analysis is primarily concerned with how real or complex-valued functions (often on very general domains) can be studied using symmetries such as <a href="/wiki/Translations" class="mw-redirect" title="Translations">translations</a> or <a href="/wiki/Rotations" class="mw-redirect" title="Rotations">rotations</a> (for instance via the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> and its relatives); this field is of course related to real-variable harmonic analysis, but is perhaps closer in spirit to <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> on <a href="/wiki/Topological_group" title="Topological group">topological groups</a>. The core motivating ideas are the various <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a>, which can be generalized to a transform of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> defined on Hausdorff <a href="/wiki/Locally_compact_group" title="Locally compact group">locally compact topological groups</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>One of the major results in the theory of functions on <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Locally_compact_group" title="Locally compact group">locally compact groups</a> is called <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a>. Harmonic analysis studies the properties of that duality. Different generalization of <a href="/wiki/Fourier_transforms" class="mw-redirect" title="Fourier transforms">Fourier transforms</a> attempts to extend those features to different settings, for instance, first to the case of general <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Topological_groups" class="mw-redirect" title="Topological groups">topological groups</a> and second to the case of non-abelian <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, the <a href="/wiki/Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This choice of harmonics enjoys some of the valuable properties of the classical Fourier transform in terms of carrying convolutions to pointwise products or otherwise showing a certain understanding of the underlying <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure. See also: <a href="/wiki/Non-commutative_harmonic_analysis" class="mw-redirect" title="Non-commutative harmonic analysis">Non-commutative harmonic analysis</a>. </p><p>If the group is neither abelian nor compact, no general satisfactory theory is currently known ("satisfactory" means at least as strong as the <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a>). However, many specific cases have been analyzed, for example, <a href="/wiki/Special_linear_group" title="Special linear group">SL<sub><i>n</i></sub></a>. In this case, <a href="/wiki/Group_representation" title="Group representation">representations</a> in infinite <a href="/wiki/Dimension_(mathematics_and_physics)" class="mw-redirect" title="Dimension (mathematics and physics)">dimensions</a> play a crucial role. </p> <div class="mw-heading mw-heading2"><h2 id="Applied_harmonic_analysis">Applied harmonic analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=4" title="Edit section: Applied harmonic analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bass_Guitar_Time_Signal_of_open_string_A_note_(55_Hz).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png/400px-Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png" decoding="async" width="400" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png/600px-Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png/800px-Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png 2x" data-file-width="1200" data-file-height="600" /></a><figcaption> Bass-guitar time signal of open-string A note (55 Hz)</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Fourier_Transform_of_bass_guitar_time_signal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fourier_Transform_of_bass_guitar_time_signal.png/400px-Fourier_Transform_of_bass_guitar_time_signal.png" decoding="async" width="400" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fourier_Transform_of_bass_guitar_time_signal.png/600px-Fourier_Transform_of_bass_guitar_time_signal.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fourier_Transform_of_bass_guitar_time_signal.png/800px-Fourier_Transform_of_bass_guitar_time_signal.png 2x" data-file-width="1200" data-file-height="600" /></a><figcaption> Fourier transform of bass-guitar time signal of open-string A note (55 Hz)<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components. Ocean <a href="/wiki/Tide" title="Tide">tides</a> and vibrating <a href="/wiki/String_(music)" title="String (music)">strings</a> are common and simple examples. The theoretical approach often tries to describe the system by a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> or <a href="/wiki/System_of_equations" title="System of equations">system of equations</a> to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components. The specific equations depend on the field, but theories generally try to select equations that represent significant principles that are applicable. </p><p>The experimental approach is usually to <a href="/wiki/Data_collection" title="Data collection">acquire data</a> that accurately quantifies the phenomenon. For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included. In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected. </p><p>For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz. The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves. The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, shown in the lower figure. There is a prominent peak at 55 Hz, but other peaks at 110 Hz, 165 Hz, and at other frequencies corresponding to integer multiples of 55 Hz. In this case, 55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as <a href="/wiki/Harmonic" title="Harmonic">harmonics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_branches">Other branches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=5" title="Edit section: Other branches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Study of the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> and <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> on <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domains</a>, <a href="/wiki/Manifold" title="Manifold">manifolds</a>, and (to a lesser extent) <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> is also considered a branch of harmonic analysis. See, e.g., <a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">hearing the shape of a drum</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>Harmonic analysis on Euclidean spaces deals with properties of the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> on <b>R</b><sup><i>n</i></sup> that have no analog on general groups. For example, the fact that the Fourier transform is rotation-invariant. Decomposing the Fourier transform into its radial and spherical components leads to topics such as <a href="/wiki/Bessel_function" title="Bessel function">Bessel functions</a> and <a href="/wiki/Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">spherical harmonics</a>.</li> <li>Harmonic analysis on tube domains is concerned with generalizing properties of <a href="/wiki/Hardy_space" title="Hardy space">Hardy spaces</a> to higher dimensions.</li> <li><a href="/wiki/Automorphic_forms" class="mw-redirect" title="Automorphic forms">Automorphic forms</a> are generalized harmonic functions, with respect to a symmetry group. They are an old and at the same time active area of development in harmonic analysis due to their connections to the <a href="/wiki/Langlands_program" title="Langlands program">Langlands program</a>.</li> <li>Non linear harmonic analysis is the use of harmonic and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> tools and techniques to study <a href="/wiki/Nonlinear_systems" class="mw-redirect" title="Nonlinear systems">nonlinear systems</a>. This includes both problems with infinite <a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a> and also non linear <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operators</a> and <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">equations</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Major_results">Major results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=6" title="Edit section: Major results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Harmonic_analysis&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> for computing periodicity in evenly-spaced data</li> <li><a href="/wiki/Harmonic_(mathematics)" title="Harmonic (mathematics)">Harmonic (mathematics)</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a> for computing periodicity in unevenly spaced data</li> <li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Tate%27s_thesis" title="Tate's thesis">Tate's thesis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.etymonline.com/index.php?term=harmonic">"harmonic"</a>. <i><a href="/wiki/Online_Etymology_Dictionary" title="Online Etymology Dictionary">Online Etymology Dictionary</a></i>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf">https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Bare_URLs" title="Wikipedia:Bare URLs"><span title="A full citation of this PDF document is required to prevent link rot. (August 2024)">bare URL PDF</span></a></i>]</sup></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFN._Vilenkin1968" class="citation book cs1">N. Vilenkin (1968). <i>Special functions and the theory of group representation</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Special+functions+and+the+theory+of+group+representation&rft.date=1968&rft.au=N.+Vilenkin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Atiyah-Singer_index_theorem" class="mw-redirect" title="Atiyah-Singer index theorem">Atiyah-Singer index theorem</a></div></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/harmonic-analysis">"Harmonic analysis | Mathematics, Fourier Series & Waveforms | Britannica"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Harmonic+analysis+%26%23124%3B+Mathematics%2C+Fourier+Series+%26+Waveforms+%26%23124%3B+Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fharmonic-analysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf">https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Bare_URLs" title="Wikipedia:Bare URLs"><span title="A full citation of this PDF document is required to prevent link rot. (August 2024)">bare URL PDF</span></a></i>]</sup></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf">https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Bare_URLs" title="Wikipedia:Bare URLs"><span title="A full citation of this PDF document is required to prevent link rot. (August 2024)">bare URL PDF</span></a></i>]</sup></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlain_Robert" class="citation book cs1">Alain Robert. <i>Introduction to the Representation Theory of Compact and Locally Compact Groups</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Representation+Theory+of+Compact+and+Locally+Compact+Groups&rft.au=Alain+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGerald_B_Folland" class="citation book cs1">Gerald B Folland. <i>A Course in Abstract Harmonic Analysis</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Abstract+Harmonic+Analysis&rft.au=Gerald+B+Folland&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlain_Robert" class="citation book cs1">Alain Robert. <i>Introduction to the Representation Theory of Compact and Locally Compact Groups</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Representation+Theory+of+Compact+and+Locally+Compact+Groups&rft.au=Alain+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://sourceforge.net/projects/amoreaccuratefouriertransform/">"A More Accurate Fourier Transform"</a>. <i>SourceForge</i>. 2015-07-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SourceForge&rft.atitle=A+More+Accurate+Fourier+Transform&rft.date=2015-07-07&rft_id=https%3A%2F%2Fsourceforge.net%2Fprojects%2Famoreaccuratefouriertransform%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTerras2013" class="citation book cs1">Terras, Audrey (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LcHBAAAAQBAJ&q=harmonic+analysis+hear+shape+of+a+drum&pg=PA37"><i>Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane</i></a> (2nd ed.). New York, NY: Springer. p. 37. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1461479710" title="Special:BookSources/978-1461479710"><bdi>978-1461479710</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">12 December</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonic+Analysis+on+Symmetric+Spaces-Euclidean+Space%2C+the+Sphere%2C+and+the+Poincar%C3%A9+Upper+Half-Plane&rft.place=New+York%2C+NY&rft.pages=37&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft.isbn=978-1461479710&rft.aulast=Terras&rft.aufirst=Audrey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLcHBAAAAQBAJ%26q%3Dharmonic%2Banalysis%2Bhear%2Bshape%2Bof%2Ba%2Bdrum%26pg%3DPA37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoifmanMeyer1987" class="citation book cs1">Coifman, R. R.; Meyer, Yves (1987). <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9781400882090-002/html?lang=en">"Non-Linear Harmonic Analysis, Operator Theory and P.d.e."</a>. <i>Beijing Lectures in Harmonic Analysis. (AM-112)</i>. pp. 1–46. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400882090-002">10.1515/9781400882090-002</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-8209-0" title="Special:BookSources/978-1-4008-8209-0"><bdi>978-1-4008-8209-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Non-Linear+Harmonic+Analysis%2C+Operator+Theory+and+P.d.e.&rft.btitle=Beijing+Lectures+in+Harmonic+Analysis.+%28AM-112%29&rft.pages=1-46&rft.date=1987&rft_id=info%3Adoi%2F10.1515%2F9781400882090-002&rft.isbn=978-1-4008-8209-0&rft.aulast=Coifman&rft.aufirst=R.+R.&rft.au=Meyer%2C+Yves&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdocument%2Fdoi%2F10.1515%2F9781400882090-002%2Fhtml%3Flang%3Den&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic+analysis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=9" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a> and <a href="/wiki/Guido_Weiss" title="Guido Weiss">Guido Weiss</a>, <i>Introduction to Fourier Analysis on Euclidean Spaces</i>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, 1971. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08078-X" title="Special:BookSources/0-691-08078-X">0-691-08078-X</a></li> <li><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a> with Timothy S. Murphy, <i>Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals</i>, Princeton University Press, 1993.</li> <li><a href="/wiki/Elias_M._Stein" title="Elias M. Stein">Elias Stein</a>, <i>Topics in Harmonic Analysis Related to the Littlewood-Paley Theory</i>, Princeton University Press, 1970.</li> <li><a href="/wiki/Yitzhak_Katznelson" title="Yitzhak Katznelson">Yitzhak Katznelson</a>, <i>An introduction to harmonic analysis</i>, Third edition. Cambridge University Press, 2004. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83829-0" title="Special:BookSources/0-521-83829-0">0-521-83829-0</a>; 0-521-54359-2</li> <li><a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a>, <a rel="nofollow" class="external text" href="https://www.math.ucla.edu/~tao/preprints/fourier.pdf">Fourier Transform</a>. (Introduces the decomposition of functions into odd + even parts as a harmonic decomposition over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}"></span>.)</li> <li>Yurii I. Lyubich. <i>Introduction to the Theory of Banach Representations of Groups</i>. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.</li> <li><a href="/wiki/George_W._Mackey" class="mw-redirect" title="George W. Mackey">George W. Mackey</a>, <a rel="nofollow" class="external text" href="https://doi.org/10.1090/S0273-0979-1980-14783-7">Harmonic analysis as the exploitation of symmetry–a historical survey</a>, <i>Bull. Amer. Math. Soc.</i> 3 (1980), 543–698.</li> <li>M. Bujosa, A. Bujosa and A. Garcıa-Ferrer. <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109/TSP.2015.2469640">Mathematical Framework for Pseudo-Spectra of Linear Stochastic Difference Equations</a>, <i>IEEE Transactions on Signal Processing</i> vol. 63 (2015), 6498–6509.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic_analysis&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output 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