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<title>Fourier-Transformation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Fourier-Transformation</h1>

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Die Fourier-Transformation (genauer die kontinuierliche Fourier-Transformation; Aussprache: <style data-mw-deduplicate="TemplateStyles:r227981795">

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</style><a href="Liste_der_IPA-Zeichen" title="Liste der IPA-Zeichen">[fuʁie]</a>) ist eine mathematische Methode aus dem Bereich der <a href="Fourier-Analysis" title="Fourier-Analysis">Fourier-Analyse</a>, mit der <a href="Aperiodisch" title="Aperiodisch">aperiodische</a> Signale in ein kontinuierliches <a href="Frequenzspektrum" title="Frequenzspektrum">Spektrum</a> zerlegt werden. Die Funktion, die dieses Spektrum beschreibt, nennt man auch Fourier-Transformierte oder Spektralfunktion. Es handelt sich dabei um eine <a href="Integraltransformation" class="mw-redirect" title="Integraltransformation">Integraltransformation</a>, die nach dem Mathematiker <a href="Jean_Baptiste_Joseph_Fourier" class="mw-redirect" title="Jean Baptiste Joseph Fourier">Jean Baptiste Joseph Fourier</a> benannt ist. Fourier führte im Jahr 1822 die <a href="Fourier-Reihe" class="mw-redirect" title="Fourier-Reihe">Fourier-Reihe</a> ein, die jedoch nur für periodische Signale definiert ist und zu einem diskreten Frequenzspektrum führt.

Es gibt einige Anwendungsfälle, in denen die Fourier-Transformation mittels eines Computers berechnet werden soll. Dafür wurde die <a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">Diskrete Fourier-Transformation</a> beziehungsweise die <a href="Schnelle_Fourier-Transformation" title="Schnelle Fourier-Transformation">Schnelle Fourier-Transformation</a> eingeführt.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d18f6a375b29725466fd705ddd852c78b6969a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.462ex; height:3.176ex;" alt="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}" loading="lazy"> eine integrierbare Funktion, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74c288d1089f1ec85b01b4de25c441fc792bd2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}" loading="lazy"> den <a href="Lp-Raum" title="Lp-Raum">Lebesgue-Raum</a> bezeichnet. Die (kontinuierliche) Fourier-Transformierte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}f}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5dcde8694af884309e77803facff81e35836c72.svg" 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}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"> ist definiert durch

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}f)(y)={\frac {1}{{\sqrt {2\pi }}^{\ n}}}\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} y\cdot x}\,\mathrm {d} x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c6381d110b482bb74d3c7d495bc07850b3a88d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.623ex; height:6.176ex;" alt="{\displaystyle ({\mathcal {F}}f)(y)={\frac {1}{{\sqrt {2\pi }}^{\ n}}}\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} y\cdot x}\,\mathrm {d} x}" loading="lazy"></dd></dl>

und die zugehörige inverse Transformation lautet:

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}^{\ n}}}\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} y\cdot x}\,\mathrm {d} y\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d139b5deb249762ec11068ccc7daa81b26f44aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.075ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}^{\ n}}}\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} y\cdot x}\,\mathrm {d} y\;.}" loading="lazy"></dd></dl>

Dabei gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b637d7a6d64d797391d40ceae9e8696e5d76f15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} y}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40ec4b0f294f77937ccbd41e91214ba60b67ac65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} y}" loading="lazy"> sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale <a href="Volumenelement" class="mw-redirect" title="Volumenelement">Volumenelemente</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\cdot x}">

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<annotation encoding="application/x-tex">{\displaystyle y\cdot x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d72f095008edb2d2cdc54c116b0edfeefdbe6486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.164ex; height:2.009ex;" alt="{\displaystyle y\cdot x}" loading="lazy"> das <a href="Standardskalarprodukt" title="Standardskalarprodukt">Standardskalarprodukt</a> der Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

Die Normierungskonstante ist in der Literatur nicht einheitlich. In der Theorie der Pseudodifferentialoperatoren und in der Signalverarbeitung ist es üblich, den Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(2\pi )^{n/2}}">

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<annotation encoding="application/x-tex">{\displaystyle 1/(2\pi )^{n/2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/055cbe831e83426f2e767c795dabfdff3699fa05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.491ex; height:3.343ex;" alt="{\displaystyle 1/(2\pi )^{n/2}}" loading="lazy"> in der Transformation wegzulassen, sodass stattdessen die Rücktransformation den Vorfaktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(2\pi )^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle 1/(2\pi )^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c80e2150249bbb7ab2e931df0a195a11dd65ea7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.847ex; height:2.843ex;" alt="{\displaystyle 1/(2\pi )^{n}}" loading="lazy"> erhält. Die Transformation lautet dann:

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}f)(y)=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} y\cdot x}\,\mathrm {d} x\;,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a78b4b52f783c88bbe3d407c60bd3b09bceba98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.462ex; height:5.676ex;" alt="{\displaystyle ({\mathcal {F}}f)(y)=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} y\cdot x}\,\mathrm {d} x\;,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} y\cdot x}\,\mathrm {d} y\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9be915c0a77b83b1e515bb7b86736e3865c330e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.368ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} y\cdot x}\,\mathrm {d} y\;.}" loading="lazy"></dd></dl>

Hier taucht ein Vorfaktor auf, so dass die Anwendung des <a href="Satz_von_Plancherel" title="Satz von Plancherel">Satzes von Plancherel</a> nicht direkt möglich ist, weil die Fouriertransformation dann keine <a href="Unit%C3%A4re_Abbildung" title="Unitäre Abbildung">unitäre Abbildung</a> mehr auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} ^{n})\cap L^{2}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} ^{n})\cap L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebadc21323e782c4971b93b9dd10841fde46ece5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.268ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} ^{n})\cap L^{2}(\mathbb {R} ^{n})}" loading="lazy"> ist und so die <a href="Signalleistung" class="mw-redirect" title="Signalleistung">Signalleistung</a> ändert. Dies kann jedoch (wie bei allen Orthogonaltransformationen) einfach durch eine Substitution (Reskalierung der Abszisse) ausgeglichen werden und stellt damit kein grundlegendes Problem dar. Genau dies wird in der Literatur zu Signalverarbeitung und Systemtheorie vorgeschlagen, indem von der natürlichen Frequenz auf die Kreisfrequenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi y}">

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<annotation encoding="application/x-tex">{\displaystyle \omega =2\pi y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/934af3394efee69b402786f5b63cdb75d83b635d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.194ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi y}" loading="lazy"> (die den Faktor beinhaltet) übergegangen wird:

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}f)(y)=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-2\pi \mathrm {i} y\cdot x}\,\mathrm {d} x\;=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} \omega \cdot x}\,\mathrm {d} x\;}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab3b3e5bea2883258c2489c565a167be129f49b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.621ex; height:5.676ex;" alt="{\displaystyle ({\mathcal {F}}f)(y)=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-2\pi \mathrm {i} y\cdot x}\,\mathrm {d} x\;=\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {e} ^{-\mathrm {i} \omega \cdot x}\,\mathrm {d} x\;}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{2\pi \mathrm {i} y\cdot x}\,\mathrm {d} y\;=\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} \omega \cdot x}\,\mathrm {d} y\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786dcd05e954f314969887dd16d97e01c1f95854.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.15ex; height:5.676ex;" alt="{\displaystyle f(x)=\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{2\pi \mathrm {i} y\cdot x}\,\mathrm {d} y\;=\int _{\mathbb {R} ^{n}}({\mathcal {F}}f)(y)\,\mathrm {e} ^{\mathrm {i} \omega \cdot x}\,\mathrm {d} y\;.}" loading="lazy"></dd></dl>

Die reelle Form der Fourier-Transformation wird als <a href="Hartley-Transformation" title="Hartley-Transformation">Hartley-Transformation</a> bezeichnet. Für reelle Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"> kann die Fourier-Transformation durch die <a href="Sinus-_und_Kosinus-Transformation" title="Sinus- und Kosinus-Transformation">Sinus- und Kosinus-Transformation</a> substituiert werden.

<div class="mw-heading mw-heading2"><h2 id="Anwendungsfälle">Anwendungsfälle</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Kompressionsverfahren_für_die_digitale_Kommunikation">Kompressionsverfahren für die digitale Kommunikation</h3></div>

Die Kompression von digitalen Daten auf Basis der Fourier-Transformation ist eine zentrale Technologie für Kommunikation, Datenaustausch und <a href="Streaming_Media" title="Streaming Media">Streaming</a> von Medien im (mobilen) Internet.<a href="#cite_note-1">[1]</a>

Beispielsweise wird zur Kompression von Audio-Daten (etwa um eine <a href="MP3" title="MP3">MP3</a> Datei zu erzeugen) das Audio-Signal in den Frequenz-Raum transformiert. Die Transformation erfolgt über das Verfahren der <a href="Modifizierte_diskrete_Kosinustransformation" title="Modifizierte diskrete Kosinustransformation">(modifizierten) diskreten Kosinustransformation</a>, welches der <a href="Schnelle_Fourier-Transformation" title="Schnelle Fourier-Transformation">schnellen Fourier-Transformation</a> ähnelt. Im Frequenzraum werden dann alle Frequenzen, die Menschen nicht hören können oder die nur wenig zum subjektiven Empfinden des Klangs beitragen, entfernt. Das Ergebnis wird im letzten Schritt aus dem Frequenz-Raum <a class="mw-selflink-fragment" href="#Rücktransformationsformel">rücktransformiert</a> – daraus erhält man, auf Grund des verringerten Frequenzumfangs, eine deutlich kleinere (komprimierte) Audio-Datei.<a href="#cite_note-2">[2]</a>

In vergleichbaren Verfahren können Bilder (<a href="JPEG" title="JPEG">JPEG</a> Kompression) oder Filme (<a href="MPEG-4" title="MPEG-4">MPEG-4</a>) komprimiert werden.

<div class="mw-heading mw-heading3"><h3 id="Signalanalyse">Signalanalyse</h3></div>

In der <a href="Signalanalyse" title="Signalanalyse">Signalanalyse</a> werden mittels Fourier-Transformation <a href="Frequenzanalyse" class="mw-disambig" title="Frequenzanalyse">Frequenzanalysen</a> von Signalen durchgeführt. Hierzu wird das Verfahren der <a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">diskreten Fourier-Transformation</a> bzw. der <a href="Schnelle_Fourier-Transformation" title="Schnelle Fourier-Transformation">schnellen Fourier-Transformation</a> genutzt. Ein Beispiel für die <a href="Schnelle_Fourier-Transformation#Anwendungen" title="Schnelle Fourier-Transformation">Vielzahl von technischen Anwendungen</a> ist die Nutzung der Signalanalyse bei der Erstellung von Bildern mittels <a href="Magnetresonanztomographie" title="Magnetresonanztomographie">Magnetresonanztomographie</a>.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_Signalanalyse_in_der_Akustik">Beispiel Signalanalyse in der Akustik</h3></div>

Der reine <a href="Kammerton" title="Kammerton">Kammerton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}}" loading="lazy"> ist eine Sinuswelle mit der Frequenz 440 Hz, also 440 Schwingungen pro Sekunde. Eine ideale Stimmgabel gibt genau dieses Sinussignal ab. Der gleiche Ton gespielt mit einem anderen Musikinstrument (nicht-ideale Stimmgabel), ist eine Zusammensetzung/Überlagerung aus Wellen verschiedener Wellenlängen. Diese sind bezüglich ihrer Frequenz normalerweise ganzzahlige Vielfache der Frequenz des Grundtons. Die Zusammensetzung und jeweilige Amplitude dieser Wellen ist bestimmend für die <a href="Klangfarbe" title="Klangfarbe">Klangfarbe</a> jedes Musikinstruments. Nur die Welle mit der größten Wellenlänge, der <a href="Grundfrequenz" title="Grundfrequenz">Grundton</a> des Signals, hat dabei die Frequenz 440 Hz. Die anderen Wellen, die <a href="Harmonische" title="Harmonische">Obertöne</a>, haben höhere Frequenzen.

An der Fourier-Transformierten des Tonsignals kann man direkt die verschiedenen Frequenzen/Wellenlängen der Wellenzusammensetzung ablesen. Diese Eigenschaft kann man beispielsweise für die automatische Erkennung von Tonhöhen und Musikinstrumenten in einem Tonsignal ausnutzen.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Wellenpaket_im_Zeit-_und_Frequenzbereich">Wellenpaket im Zeit- und Frequenzbereich</h3></div>

Gegeben sei ein Signalimpuls als Summe zweier Cosinus-Funktionen mit Frequenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}=2{,}5\,\mathrm {Hz} }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}=2{,}5\,\mathrm {Hz} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12f0d555e79fbae6bdbea71d1defc8ada078c960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.426ex; height:2.509ex;" alt="{\displaystyle f_{1}=2{,}5\,\mathrm {Hz} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}=7\,\mathrm {Hz} ,}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}=7\,\mathrm {Hz} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e258212d166c136cc5a02d6d7b039c5528a5255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.264ex; height:2.509ex;" alt="{\displaystyle f_{2}=7\,\mathrm {Hz} ,}" loading="lazy"> multipliziert mit einer <a href="Normalverteilung" title="Normalverteilung">Gauß-Glocke</a> der Breite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}=(2\pi )^{-0,5}\,\mathrm {s} \approx 0{,}4\,\mathrm {s} }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{t}=(2\pi )^{-0,5}\,\mathrm {s} \approx 0{,}4\,\mathrm {s} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9bf75dd56a2d7daf47df3ccfd1b5f78f11c4716.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.845ex; height:3.176ex;" alt="{\displaystyle \sigma _{t}=(2\pi )^{-0,5}\,\mathrm {s} \approx 0{,}4\,\mathrm {s} }" loading="lazy">:

<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)=(4\cos 2\pi f_{1}t+2\cos 2\pi f_{2}t)\,\mathrm {e} ^{-0,5(t/\sigma _{t})^{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0de19102c1cca3db0b509aae059fc46db8be9d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.846ex; height:3.676ex;" alt="{\displaystyle y(t)=(4\cos 2\pi f_{1}t+2\cos 2\pi f_{2}t)\,\mathrm {e} ^{-0,5(t/\sigma _{t})^{2}}}" loading="lazy"></dd></dl>

Die spezielle Breite wurde gewählt, weil damit der Exponentialterm ohne weiteren Vorfaktor die Fläche 1 s hat. Er hat somit auch die Amplitude 1, sodass der Funktionswert bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> die Summe der Kosinus-Amplituden ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4+2=6)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (4+2=6)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d090741637915d678af568d5e68ec4221fb956b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.236ex; height:2.843ex;" alt="{\displaystyle (4+2=6)}" loading="lazy">.

Das obere Teildiagramm zeigt den Funktionsgraphen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/397de1edef5bf2ee15c020f325d7d781a3aa7f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" loading="lazy">. Darunter dargestellt ist das Amplitudenspektrum, also der Betrag der Fourier-Transformierten, die eine komplexwertige Funktion der Frequenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"> ist. Es besteht aus vier gaußförmigen Spektrallinien bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm f_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm f_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85ab8131faa5fa664a6b1c35ec4e7d8d98fdae3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.002ex; height:2.509ex;" alt="{\displaystyle \pm f_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm f_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm f_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8695206933f2a4ff944165882d5b11645f1701e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.002ex; height:2.509ex;" alt="{\displaystyle \pm f_{2}}" loading="lazy">. Die Linienbreite beträgt jeweils <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{f}=(2\pi )^{-0,5}\,\mathrm {Hz} \approx 0{,}4\,\mathrm {Hz} }">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{f}=(2\pi )^{-0,5}\,\mathrm {Hz} \approx 0{,}4\,\mathrm {Hz} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8db7451c2b92ff2c4a62a1b0d76fee9f5ae320cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.873ex; height:3.343ex;" alt="{\displaystyle \sigma _{f}=(2\pi )^{-0,5}\,\mathrm {Hz} \approx 0{,}4\,\mathrm {Hz} }" loading="lazy">. Ein breiteres Wellenpaket würde zu schmaleren Spektrallinien führen. Das Produkt der Breiten im Zeit- und Frequenzbereich ist dimensionslos und beträgt für gaußförmige Hüllkurven stets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}\sigma _{f}=(2\pi )^{-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{t}\sigma _{f}=(2\pi )^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a7ae0971f95361021cd04c6b7ae8289c0d8b4dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.352ex; height:3.343ex;" alt="{\displaystyle \sigma _{t}\sigma _{f}=(2\pi )^{-1}}" loading="lazy">, eine Art <a href="Unsch%C3%A4rferelation" class="mw-redirect" title="Unschärferelation">Unschärferelation</a>. Das Produkt wäre 1 bei Verwendung der Kreisfrequenz.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> eine <a href="Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">physikalische Größe</a> ist, was bedeuten dann die Werte des Amplitudenspektrums? Für die Achsbeschriftung der Diagramme wurde vereinfacht angenommen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/888453430f9613fcab7a4ec7a3dfd75260d51427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.215ex; height:3.009ex;" alt="{\displaystyle y^{2}}" loading="lazy"> als <a href="Leistungsgr%C3%B6%C3%9Fe" title="Leistungsgröße">Leistungsgröße</a> direkt die Einheit <a href="Watt_(Einheit)" title="Watt (Einheit)">Watt</a> hat, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> als sog. Feldgröße also die Einheit √Watt. Damit beträgt die mittlere Leistung der beiden Kosinusfunktionen 0,5·42 bzw. 0,5·22 Watt, zusammen 10 Watt, zu multiplizieren mit der Fläche der quadrierten Hüllkurve. Die Hüllkurve selbst ist normiert, Fläche 1 s. Ihr Quadrat ist wieder eine Gauß-Glocke, hat gleiche Höhe (1), aber halbe Varianz, Fläche 1 s/√2. Damit beträgt die Energie des Wellenpakets etwa 7,07 <a href="Joule" title="Joule">Joule</a>. Die numerische Integration, siehe den Python-Code auf der Bildbeschreibungsseite, liefert den gleichen Wert, auch für das Energiespektrum (Quadrat des Amplitudenspektrums). Die Werte des Amplitudenspektrums haben folglich die Einheit √Joule pro Hz. Da die Zahlenwerte der Breiten im Zeit- und Frequenzraum (nicht zufällig) gleich sind und die Zahl der Komponenten sich verdoppelt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pm f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\pm f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cf7d46da7637844d28aaecac7d215f13dbf66e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.896ex; height:2.843ex;" alt="{\displaystyle (\pm f)}" loading="lazy">, halbieren sich die Zahlenwerte der Amplituden (Linienhöhen).

<div class="mw-heading mw-heading3"><h3 id="Beispielhafte_Berechnung_einer_Fourier-Transformierten">Beispielhafte Berechnung einer Fourier-Transformierten</h3></div>

Es soll das Frequenzspektrum einer gedämpften Schwingung mit ausreichend schwacher Dämpfung untersucht werden. Diese kann durch folgende Funktion beschrieben werden:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=x_{0}\cdot \mathrm {e} ^{-{\frac {t}{\tau }}}\cdot \cos(\omega _{\rm {s}}t)\Theta (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdd3eb7654762025e5451f475e730ec0296755ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.38ex; height:3.843ex;" alt="{\displaystyle f(t)=x_{0}\cdot \mathrm {e} ^{-{\frac {t}{\tau }}}\cdot \cos(\omega _{\rm {s}}t)\Theta (t)}" loading="lazy"></dd></dl>

oder in komplexer Schreibweise:

<msub>

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<msup>

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<mfrac>

</mfrac>

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</msup>

<mfrac>

</mfrac>

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<msup>

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<msub>

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</msub>

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</msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=x_{0}\cdot \mathrm {e} ^{-{\frac {t}{\tau }}}\cdot {\tfrac {1}{2}}(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t})\Theta (t)\;.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae0db6f8fb2f3cb502bbc8af3bf812362629c6b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.229ex; height:4.176ex;" alt="{\displaystyle f(t)=x_{0}\cdot \mathrm {e} ^{-{\frac {t}{\tau }}}\cdot {\tfrac {1}{2}}(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t})\Theta (t)\;.}" loading="lazy"></dd></dl>

Hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> die <a href="Amplitude" title="Amplitude">Amplitude</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\rm {s}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\rm {s}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0152689e3fffd596a8846cb2309fe03a000aa68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\displaystyle \omega _{\rm {s}}}" loading="lazy"> die <a href="Kreisfrequenz" title="Kreisfrequenz">Kreisfrequenz</a> der Schwingung, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" 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 }" loading="lazy"> die Zeit, in der die Amplitude um den Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/e}">

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<annotation encoding="application/x-tex">{\displaystyle 1/e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3f5e6342b5d465a51daf323f383ed1c8e51f43f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle 1/e}" loading="lazy"> abfällt, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/415d180b65fdc75e95cb50e5786e6c56989dd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.457ex; height:2.843ex;" alt="{\displaystyle \Theta (t)}" loading="lazy"> die <a href="Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a>.

Das heißt, die Funktion ist nur für positive Zeiten nicht null.

Man erhält

<mtr>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(\omega )=({\mathcal {F}}f)(\omega )&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x_{0}\cdot \mathrm {e} ^{-t/\tau }\cdot {\tfrac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t}\right)\Theta (t)\cdot \mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {x_{0}}{\sqrt {2\pi }}}\int _{0}^{\infty }\mathrm {e} ^{-t/\tau }\cdot {\tfrac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t}\right)\cdot \mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\int _{0}^{\infty }\left(\mathrm {e} ^{-t\left(1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )\right)}+\mathrm {e} ^{-t\left(1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )\right)}\right)\,\mathrm {d} t\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\left[-{\frac {1}{1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )}}\mathrm {e} ^{-t\left(1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )\right)}-{\frac {1}{1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )}}\mathrm {e} ^{-t\left(1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )\right)}\right]_{0}^{\infty }\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\left({\frac {1}{1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )}}+{\frac {1}{1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )}}\right)\\&={\frac {x_{0}}{\sqrt {2\pi }}}{\frac {1/\tau +\mathrm {i} \omega }{(1/\tau +\mathrm {i} \omega )^{2}+\omega _{\rm {s}}^{2}}}\,.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26f20b97dc90cba9a1e81d484f03fbeda4935f02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.338ex; width:96.043ex; height:45.843ex;" alt="{\displaystyle {\begin{aligned}F(\omega )=({\mathcal {F}}f)(\omega )&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x_{0}\cdot \mathrm {e} ^{-t/\tau }\cdot {\tfrac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t}\right)\Theta (t)\cdot \mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {x_{0}}{\sqrt {2\pi }}}\int _{0}^{\infty }\mathrm {e} ^{-t/\tau }\cdot {\tfrac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} \omega _{\rm {s}}t}+\mathrm {e} ^{-\mathrm {i} \omega _{\rm {s}}t}\right)\cdot \mathrm {e} ^{-\mathrm {i} \omega t}\,\mathrm {d} t\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\int _{0}^{\infty }\left(\mathrm {e} ^{-t\left(1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )\right)}+\mathrm {e} ^{-t\left(1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )\right)}\right)\,\mathrm {d} t\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\left[-{\frac {1}{1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )}}\mathrm {e} ^{-t\left(1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )\right)}-{\frac {1}{1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )}}\mathrm {e} ^{-t\left(1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )\right)}\right]_{0}^{\infty }\\&={\frac {x_{0}}{2{\sqrt {2\pi }}}}\left({\frac {1}{1/\tau -\mathrm {i} (\omega _{\rm {s}}-\omega )}}+{\frac {1}{1/\tau +\mathrm {i} (\omega _{\rm {s}}+\omega )}}\right)\\&={\frac {x_{0}}{\sqrt {2\pi }}}{\frac {1/\tau +\mathrm {i} \omega }{(1/\tau +\mathrm {i} \omega )^{2}+\omega _{\rm {s}}^{2}}}\,.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Linearität">Linearität</h3></div>

Die Fourier-Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" 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}}}" loading="lazy"> ist ein <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a>. Das heißt, es gilt

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(a\cdot f+b\cdot g)=a\cdot {\mathcal {F}}(f)+b\cdot {\mathcal {F}}(g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f8607386be8c8f1c9dd39ff36e23a02d27191a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.947ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(a\cdot f+b\cdot g)=a\cdot {\mathcal {F}}(f)+b\cdot {\mathcal {F}}(g)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Stetigkeit">Stetigkeit</h3></div>

Die Fourier-Transformation ist ein stetiger Operator vom <a href="Lp-Raum" title="Lp-Raum">Raum der integrierbaren Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} ^{n})}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d919ffb3de31d39cecd5f28a2992e96b37dcb9d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} ^{n})}" loading="lazy"> in den <a href="C0-Funktion" title="C0-Funktion">Raum der Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle C_{0}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c8020df3bfa82df1eee4d5c0cd672f11c22e89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.422ex; height:2.843ex;" alt="{\displaystyle C_{0}(\mathbb {R} ^{n})}" loading="lazy">, die im Unendlichen verschwinden</a>. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle C_{0}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c8020df3bfa82df1eee4d5c0cd672f11c22e89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.422ex; height:2.843ex;" alt="{\displaystyle C_{0}(\mathbb {R} ^{n})}" loading="lazy"> ist die Menge der stetigen Funktionen bezeichnet, welche für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle |x|\to \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8c723844d0ddbc29a26e3d70a8c1b1acfeb6e0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle |x|\to \infty }" loading="lazy"> verschwinden. Die Tatsache, dass die Fourier-Transformierten im Unendlichen verschwinden, ist auch als <a href="Lemma_von_Riemann-Lebesgue" title="Lemma von Riemann-Lebesgue">Lemma von Riemann-Lebesgue</a> bekannt. Außerdem gilt die Ungleichung

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<annotation encoding="application/x-tex">{\displaystyle \|{\mathcal {F}}f\|_{L^{\infty }(\mathbb {R} ^{n})}\leq {\frac {1}{({\sqrt {2\pi }})^{n}}}\|f\|_{L^{1}(\mathbb {R} ^{n})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f1f4f9f73cd5992308a7ca88f5169182788d907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.421ex; height:6.343ex;" alt="{\displaystyle \|{\mathcal {F}}f\|_{L^{\infty }(\mathbb {R} ^{n})}\leq {\frac {1}{({\sqrt {2\pi }})^{n}}}\|f\|_{L^{1}(\mathbb {R} ^{n})}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Differentiationsregeln">Differentiationsregeln</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in S(\mathbb {R} ^{n})\subset L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in S(\mathbb {R} ^{n})\subset L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b6905c7989911e41e18794521374976d39e95e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.766ex; height:3.176ex;" alt="{\displaystyle f\in S(\mathbb {R} ^{n})\subset L^{1}(\mathbb {R} ^{n})}" loading="lazy"> eine <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Funktion</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {N} _{0}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {N} _{0}^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4323681a6fd393e31e678996352f9162ea5a91e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.225ex; height:2.843ex;" alt="{\displaystyle \alpha \in \mathbb {N} _{0}^{n}}" loading="lazy"> ein <a href="Multiindex" title="Multiindex">Multiindex</a>. Dann gilt

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}f\in S(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/071159a113c67c88d0764ec81b89b94f96d7ed7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.251ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}f\in S(\mathbb {R} ^{n})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\alpha }({\mathcal {F}}f)=(-{\rm {i}})^{|\alpha |}{\mathcal {F}}(x^{\alpha }f)}">

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<annotation encoding="application/x-tex">{\displaystyle D^{\alpha }({\mathcal {F}}f)=(-{\rm {i}})^{|\alpha |}{\mathcal {F}}(x^{\alpha }f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61978e0c96024618a51331457a7119ea32539ecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.413ex; height:3.343ex;" alt="{\displaystyle D^{\alpha }({\mathcal {F}}f)=(-{\rm {i}})^{|\alpha |}{\mathcal {F}}(x^{\alpha }f)}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}(D^{\alpha }f))(\xi )={\rm {i}}^{|\alpha |}\xi ^{\alpha }({\mathcal {F}}f)(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e2af7b7d4d0b990ef80120990c6d89448fc2dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.987ex; height:3.343ex;" alt="{\displaystyle ({\mathcal {F}}(D^{\alpha }f))(\xi )={\rm {i}}^{|\alpha |}\xi ^{\alpha }({\mathcal {F}}f)(\xi )}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Fixpunkt">Fixpunkt</h3></div>

Die Dichtefunktion

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<annotation encoding="application/x-tex">{\displaystyle \varphi (x)={\frac {1}{(2\pi )^{n/2}}}\cdot \mathrm {e} ^{-{\frac {1}{2}}\|x\|^{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb9d81369b8d4288aee8765139eab87b4218ae8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.901ex; height:6.176ex;" alt="{\displaystyle \varphi (x)={\frac {1}{(2\pi )^{n/2}}}\cdot \mathrm {e} ^{-{\frac {1}{2}}\|x\|^{2}}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" loading="lazy"> der (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionalen) Gauß’schen <a href="Normalverteilung" title="Normalverteilung">Normalverteilung</a> ist ein <a href="Fixpunkt_(Mathematik)" title="Fixpunkt (Mathematik)">Fixpunkt</a> der Fourier-Transformation. Das heißt, es gilt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" loading="lazy"> die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}\varphi )(x)=\varphi (x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5340fd893288ec3ffa8edba5963201d04f3ba72d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.153ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {F}}\varphi )(x)=\varphi (x)}" loading="lazy">.</dd></dl>

Insbesondere ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> eine <a href="Eigenfunktion" class="mw-redirect" title="Eigenfunktion">Eigenfunktion</a> der Fourier-Transformation zum <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" 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}" loading="lazy">. Mit Hilfe des <a href="Residuensatz" title="Residuensatz">Residuensatzes</a> oder mit Hilfe <a href="Partielle_Integration" title="Partielle Integration">partieller Integration</a> und Lösen einer gewöhnlichen Differentialgleichung kann in diesem Fall das Fourier-Integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\tfrac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{{\rm {i}}x\cdot \xi }\mathrm {e} ^{-{\frac {1}{2}}x^{2}}\mathrm {d} x}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\tfrac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{{\rm {i}}x\cdot \xi }\mathrm {e} ^{-{\frac {1}{2}}x^{2}}\mathrm {d} x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5bf7d02f6f8109a5d911c81148aa45a6c9a6aa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.388ex; height:5.009ex;" alt="{\displaystyle \textstyle {\tfrac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{{\rm {i}}x\cdot \xi }\mathrm {e} ^{-{\frac {1}{2}}x^{2}}\mathrm {d} x}" loading="lazy"> bestimmt werden.

<div class="mw-heading mw-heading3"><h3 id="Spiegelsymmetrie">Spiegelsymmetrie</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in S(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in S(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc292137092d22d0b844b78c60fab1897dac56e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.324ex; height:2.843ex;" alt="{\displaystyle f\in S(\mathbb {R} ^{n})}" loading="lazy"> gilt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" loading="lazy"> die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}^{2}f)(x)=({\mathcal {F}}({\mathcal {F}}f))(x)=f(-x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d34b29b0d389590eaeb67050aef6004746a3e69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.596ex; height:3.176ex;" alt="{\displaystyle ({\mathcal {F}}^{2}f)(x)=({\mathcal {F}}({\mathcal {F}}f))(x)=f(-x)}" loading="lazy">.</dd></dl>

Äquivalent lässt sich dies auf dem Schwartzraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b7c34a68f4cf179202afbf9a038399a95fd9198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.205ex; height:2.843ex;" alt="{\displaystyle S(\mathbb {R} ^{n})}" loading="lazy"> als Operatorgleichung

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{2}={\mathcal {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdcc735053879591e97f3b8c4dbfe2293c885de9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.86ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}^{2}={\mathcal {P}}}" loading="lazy"></dd></dl>

schreiben, wobei

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}:f\mapsto (x\mapsto f(-x))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80ea6a76c96e80ce8f869aa4f469f167f736eaea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.512ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}:f\mapsto (x\mapsto f(-x))}" loading="lazy"></dd></dl>

den Paritätsoperator bezeichnet.

<div class="mw-heading mw-heading3"><h3 id="Rücktransformationsformel">Rücktransformationsformel</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d18f6a375b29725466fd705ddd852c78b6969a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.462ex; height:3.176ex;" alt="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}" loading="lazy"> eine integrierbare Funktion derart, dass auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f)\in L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f)\in L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/236f771a36d07004a2a0001849d512f1341f8b43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.198ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}(f)\in L^{1}(\mathbb {R} ^{n})}" loading="lazy"> gilt. Dann gilt die Rücktransformation

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{-1}({\mathcal {F}}(f))(x)=f(x)={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} tx}{\mathcal {F}}(f)(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a3419eda2bf3b93fdc450de5b9e32be3baed8d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:50.458ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}^{-1}({\mathcal {F}}(f))(x)=f(x)={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} tx}{\mathcal {F}}(f)(t)\,\mathrm {d} t.}" loading="lazy"></dd></dl>

Diese wird auch <a href="Fouriersynthese" class="mw-redirect" title="Fouriersynthese">Fouriersynthese</a> genannt. Auf dem <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle S(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b7c34a68f4cf179202afbf9a038399a95fd9198.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.205ex; height:2.843ex;" alt="{\displaystyle S(\mathbb {R} ^{n})}" loading="lazy"> ist die Fouriertransformation ein <a href="Automorphismus" title="Automorphismus">Automorphismus</a>.

<div class="mw-heading mw-heading3"><h3 id="Faltungstheorem">Faltungstheorem</h3></div>

Das <a href="Faltungstheorem" class="mw-redirect" title="Faltungstheorem">Faltungstheorem</a> für die Fourier-Transformation besagt, dass die <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> zweier Funktionen durch die Fourier-Transformation in ihrem Bildraum in eine Multiplikation reeller Zahlen überführt wird. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f,g\in L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f301871cfd5fefc286b640307f7180071ecfdb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.612ex; height:3.176ex;" alt="{\displaystyle f,g\in L^{1}(\mathbb {R} ^{n})}" loading="lazy"> gilt also

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f*g)=(2\pi )^{\tfrac {n}{2}}\,{\mathcal {F}}(f)\cdot {\mathcal {F}}(g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df7037b87a2bb4543da02340c05f0748dbc28fb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.715ex; height:4.009ex;" alt="{\displaystyle {\mathcal {F}}(f*g)=(2\pi )^{\tfrac {n}{2}}\,{\mathcal {F}}(f)\cdot {\mathcal {F}}(g)}" loading="lazy">.</dd></dl>

Die Umkehrung des Faltungssatzes besagt<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f)*{\mathcal {F}}(g)=(2\pi )^{\tfrac {n}{2}}{\mathcal {F}}(f\cdot g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eba70fb93fe30a4ae31ad9db2cd341c94de8d06e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.327ex; height:4.009ex;" alt="{\displaystyle {\mathcal {F}}(f)*{\mathcal {F}}(g)=(2\pi )^{\tfrac {n}{2}}{\mathcal {F}}(f\cdot g)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Fourier-Transformation_von_L2-Funktionen">Fourier-Transformation von L2-Funktionen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

Für eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029ad14cb0361f43c0375039ac75a6a40046dbc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.462ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(\mathbb {R} ^{n})}" loading="lazy"> ist die Fouriertransformation mittels eines Dichtheitsargumentes definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f)(\xi )=\lim _{r\to \infty }{\frac {1}{\left(2\pi \right)^{\frac {n}{2}}}}\int _{B_{r}(0)}f(x)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8eb12fb6973a540040f723552e5afa5087945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.499ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}(f)(\xi )=\lim _{r\to \infty }{\frac {1}{\left(2\pi \right)^{\frac {n}{2}}}}\int _{B_{r}(0)}f(x)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x}" loading="lazy">.</dd></dl>

Die Konvergenz ist im Sinne von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"> zu verstehen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}(0)=\{x\in \mathbb {R} ^{n}:|x|\leq r\}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{r}(0)=\{x\in \mathbb {R} ^{n}:|x|\leq r\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ede7e3584aaa4193ef9eb8859491f748260ad14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.907ex; height:2.843ex;" alt="{\displaystyle B_{r}(0)=\{x\in \mathbb {R} ^{n}:|x|\leq r\}}" loading="lazy"> ist die Kugel um den Ursprung mit Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">. Für Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(\mathbb {R} ^{n})\cap L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(\mathbb {R} ^{n})\cap L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54b71e7f49837ed51d4f95d29c90ffecb630f541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.388ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(\mathbb {R} ^{n})\cap L^{1}(\mathbb {R} ^{n})}" loading="lazy"> stimmt diese Definition mit der aus dem ersten Abschnitt überein.

Da die Fouriertransformation bezüglich des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-Skalarproduktes unitär ist (s. u.) und

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{n})\cap L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f58983759fa2b21344652b73eba67e7e6a2cf41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.268ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{n})\cap L^{1}(\mathbb {R} ^{n})}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f15dffc817bd5f8767061d98adb57caf3519f46f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{n})}" loading="lazy"> dicht liegt, folgt,

dass die Fouriertransformation ein isometrischer Automorphismus des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f15dffc817bd5f8767061d98adb57caf3519f46f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{n})}" loading="lazy"> ist. Dies ist die Aussage des <a href="Satz_von_Plancherel" title="Satz von Plancherel">Satzes von Plancherel</a>.

<div class="mw-heading mw-heading3"><h3 id="Hausdorff-Young-Ungleichung">Hausdorff-Young-Ungleichung</h3></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq 2}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq 2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c94d35b201f4a50dd0574a25b95071e29386cde3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq 2}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}">

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0258202dfc8069365da867d2ef863a2a53f04ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.422ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}" loading="lazy">. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(\mathbb {R} ^{n})}">

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6279b562dd86e9e8d52f83c380136b41b0c9762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.467ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(\mathbb {R} ^{n})}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(f)\in L^{q}(\mathbb {R} ^{n})}">

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f)\in L^{q}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6e48f7a3c7113800efcc49e3daf7876262bd690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.132ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(f)\in L^{q}(\mathbb {R} ^{n})}" loading="lazy"> und es gilt

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<msub>

<msup>

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</msup>

<msup>

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</mrow>

</msup>

</mrow>

</msub>

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<msup>

<mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \|{\mathcal {F}}(f)\|_{L^{q}(\mathbb {R} ^{n})}\leq {\frac {1}{(2\pi )^{n\left({\frac {1}{p}}-{\frac {1}{2}}\right)}}}\|f\|_{L^{p}(\mathbb {R} ^{n})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f3b4cd9879226ecec1e26d677a0ac0d1ac53404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:37.83ex; height:7.509ex;" alt="{\displaystyle \|{\mathcal {F}}(f)\|_{L^{q}(\mathbb {R} ^{n})}\leq {\frac {1}{(2\pi )^{n\left({\frac {1}{p}}-{\frac {1}{2}}\right)}}}\|f\|_{L^{p}(\mathbb {R} ^{n})}}" loading="lazy">.</dd></dl>

Die Fourier-Transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}:L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}">

</mrow>

<msup>

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</msup>

<msup>

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}:L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49441c2daca6e18009336331b8e2bf29a592ab1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.164ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}:L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}" loading="lazy"> hat also eine <a href="Stetige_Fortsetzung" class="mw-redirect" title="Stetige Fortsetzung">Fortsetzung</a> zu einem stetigen Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}:L^{p}(\mathbb {R} ^{n})\to L^{q}(\mathbb {R} ^{n})}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

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</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}:L^{p}(\mathbb {R} ^{n})\to L^{q}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a38939acd8417acfd76a4c0fe537fc6b86c66157.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.103ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}:L^{p}(\mathbb {R} ^{n})\to L^{q}(\mathbb {R} ^{n})}" loading="lazy">, der durch

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</mrow>

</munder>

<mfrac>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

<msub>

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</mrow>

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<msup>

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</mrow>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(f)(\xi )=\lim _{r\to \infty }{\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{B_{r}(0)}f(x)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e215d4422771214cc5464fdcddfda550271f9e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.499ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}(f)(\xi )=\lim _{r\to \infty }{\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{B_{r}(0)}f(x)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x}" loading="lazy"></dd></dl>

beschrieben wird. Der Grenzwert ist hier im Sinne von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{q}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f58b2af9047d1bd68a9937af5b427b3fa4ab0a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.343ex;" alt="{\displaystyle L^{q}}" loading="lazy"> zu verstehen.

<div class="mw-heading mw-heading3"><h3 id="Differentiationsregel">Differentiationsregel</h3></div>

Falls die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"> schwach differenzierbar ist, gibt es eine Differentiationsregel analog zu denen für Schwartzfunktionen. Sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in W^{k,2}(\mathbb {R} ^{n})=H^{k}(\mathbb {R} ^{n})}">

<msup>

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<msup>

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</msup>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle f\in W^{k,2}(\mathbb {R} ^{n})=H^{k}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb5bd7c220c0193fcc792088a8ceb19ec62f7795.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.697ex; height:3.176ex;" alt="{\displaystyle f\in W^{k,2}(\mathbb {R} ^{n})=H^{k}(\mathbb {R} ^{n})}" loading="lazy"> eine <a href="Sobolew-Raum#Sobolew-Raum_(schwache_Ableitungen)" class="mw-redirect" title="Sobolew-Raum">k-mal schwach differenzierbare L2-Funktion</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> ein <a href="Multiindex" title="Multiindex">Multiindex</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |\leq k}">

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<annotation encoding="application/x-tex">{\displaystyle |\alpha |\leq k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab31d48847af82e70c82795d576d05bbf27fb109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.091ex; height:2.843ex;" alt="{\displaystyle |\alpha |\leq k}" loading="lazy">. Dann gilt

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(D^{\alpha }f)(\xi )=\mathrm {i} ^{|\alpha |}\xi ^{\alpha }{\mathcal {F}}(f)(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75e35c11d2a9fb64e07dc2444936fb2707af7340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.178ex; height:3.343ex;" alt="{\displaystyle {\mathcal {F}}(D^{\alpha }f)(\xi )=\mathrm {i} ^{|\alpha |}\xi ^{\alpha }{\mathcal {F}}(f)(\xi )}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Unitäre_Abbildung">Unitäre Abbildung</h3></div>

Die Fourier-Transformation ist bezüglich des komplexen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-Skalarproduktes ein <a href="Unit%C3%A4rer_Operator" title="Unitärer Operator">unitärer Operator</a>, das heißt, es gilt

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<msub>

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<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \langle {\mathcal {F}}(f),g\rangle _{L^{2}}=\int _{\mathbb {R} ^{n}}{\overline {{\mathcal {F}}(f)}}(x)g(x)\mathrm {d} x=\int _{\mathbb {R} ^{n}}{\overline {f}}(x){\mathcal {F}}^{-1}(g)(x)\mathrm {d} x=\langle f,{\mathcal {F}}^{-1}(g)\rangle _{L^{2}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecb68356a6093b266764db3aef27f6a579ba41dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:75.583ex; height:5.676ex;" alt="{\displaystyle \langle {\mathcal {F}}(f),g\rangle _{L^{2}}=\int _{\mathbb {R} ^{n}}{\overline {{\mathcal {F}}(f)}}(x)g(x)\mathrm {d} x=\int _{\mathbb {R} ^{n}}{\overline {f}}(x){\mathcal {F}}^{-1}(g)(x)\mathrm {d} x=\langle f,{\mathcal {F}}^{-1}(g)\rangle _{L^{2}}.}" loading="lazy"></dd></dl>

Damit liegt das <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrum</a> der Fourier-Transformation auf der <a href="Einheitskreis" title="Einheitskreis">Einheitskreislinie</a>. Im eindimensionalen Fall (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

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<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">) bilden ferner die <a href="Hermite-Funktion" class="mw-redirect" title="Hermite-Funktion">Hermite-Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(h_{n}\right)_{n\in \mathbb {N} _{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(h_{n}\right)_{n\in \mathbb {N} _{0}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1399b449b5e446eed05c3d7b0c569d84b10bf435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.7ex; height:3.343ex;" alt="{\displaystyle \left(h_{n}\right)_{n\in \mathbb {N} _{0}}}" loading="lazy"> im <a href="Lp-Raum#Der_Hilbertraum_L2" title="Lp-Raum">Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\left(\mathbb {R} \right)}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}\left(\mathbb {R} \right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2760f3b3d3b7bcce0170586909352fdc4564e857.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.512ex; height:3.176ex;" alt="{\displaystyle L^{2}\left(\mathbb {R} \right)}" loading="lazy"></a> ein <a href="Vollst%C3%A4ndiges_Orthonormalsystem" class="mw-redirect" title="Vollständiges Orthonormalsystem">vollständiges Orthonormalsystem</a> von <a href="Eigenfunktionen" class="mw-redirect" title="Eigenfunktionen">Eigenfunktionen</a> zu den Eigenwerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-\mathrm {i} \right)^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(-\mathrm {i} \right)^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c56a5efa6ec2615286cfbf58afa7bab706ca073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.483ex; height:3.009ex;" alt="{\displaystyle \left(-\mathrm {i} \right)^{n}}" loading="lazy">.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Fourier-Transformation_im_Raum_der_temperierten_Distributionen">Fourier-Transformation im Raum der temperierten Distributionen</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Temperierte_Distribution" title="Temperierte Distribution">Temperierte Distribution</a></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in S'(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle u\in S'(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3782b7e98b969215669d0df6ddd8d0891ef2a9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.083ex; height:3.009ex;" alt="{\displaystyle u\in S'(\mathbb {R} ^{n})}" loading="lazy"> eine temperierte Distribution, die Fourier-Transformierte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(u)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(u)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/735b663736d71a95370a9d8af9aa305efc39fd25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.066ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(u)}" loading="lazy"> ist für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in S(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \phi \in S(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dc97b6cdfc962400c94d58e737ea7b9ba82da85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.431ex; height:2.843ex;" alt="{\displaystyle \phi \in S(\mathbb {R} ^{n})}" loading="lazy"> definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(u)(\phi ):=u({\mathcal {F}}(\phi ))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/528ce697a91ae3877fba667dc4a3156183f62c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.266ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(u)(\phi ):=u({\mathcal {F}}(\phi ))}" loading="lazy">.</dd></dl>

Stattet man den Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle S'(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0024cb721450ee2057b7664742608b091a80b692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:3.009ex;" alt="{\displaystyle S'(\mathbb {R} ^{n})}" loading="lazy"> mit der <a href="Schwach-*-Topologie" title="Schwach-*-Topologie">Schwach-*-Topologie</a> aus, dann ist die Fourier-Transformation eine stetige, bijektive Abbildung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle S'(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0024cb721450ee2057b7664742608b091a80b692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:3.009ex;" alt="{\displaystyle S'(\mathbb {R} ^{n})}" loading="lazy">. Ihre Umkehrabbildung lautet

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<annotation encoding="application/x-tex">{\displaystyle u(\phi )(-x)={\frac {1}{(2\pi )^{n}}}{\mathcal {F}}({\mathcal {F}}(u))(\phi )(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/719a7d56303b45d2501dbfa5c12dde979f9d9ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.063ex; height:6.009ex;" alt="{\displaystyle u(\phi )(-x)={\frac {1}{(2\pi )^{n}}}{\mathcal {F}}({\mathcal {F}}(u))(\phi )(x)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Fourier-Transformation_von_Maßen">Fourier-Transformation von Maßen</h2></div>

Die Fourier-Transformation wird allgemein für endliche <a href="Borel-Ma%C3%9F" class="mw-redirect" title="Borel-Maß">Borel-Maße</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" 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}}" loading="lazy"> definiert:

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<annotation encoding="application/x-tex">{\displaystyle {\check {\mu }}(x)=\int \mathrm {e} ^{\mathrm {i} xy}\mu (\mathrm {d} y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c6220efa05da26916e067c692291cc173ed68c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.358ex; height:5.676ex;" alt="{\displaystyle {\check {\mu }}(x)=\int \mathrm {e} ^{\mathrm {i} xy}\mu (\mathrm {d} y)}" loading="lazy"></dd></dl>

heißt inverse Fourier-Transformierte des Maßes. Die <a href="Charakteristische_Funktion_(Stochastik)" title="Charakteristische Funktion (Stochastik)">charakteristische Funktion</a> ist dann die inverse Fourier-Transformierte einer <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a>.

<div class="mw-heading mw-heading2"><h2 id="Partielle_Differentialgleichungen">Partielle Differentialgleichungen</h2></div>

In der <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">Theorie der partiellen Differentialgleichungen</a> spielt die Fourier-Transformation eine wichtige Rolle. Mit ihrer Hilfe kann man Lösungen bestimmter Differentialgleichungen finden. Die <a href="#Differentiationsregel">Differentiationsregel</a> und das <a href="Faltungstheorem" class="mw-redirect" title="Faltungstheorem">Faltungstheorem</a> sind dabei von essentieller Bedeutung.

Am Beispiel der <a href="W%C3%A4rmeleitungsgleichung" title="Wärmeleitungsgleichung">Wärmeleitungsgleichung</a> wird nun gezeigt, wie man mit der Fourier-Transformation eine partielle Differentialgleichung löst. Das Anfangswertproblem der Wärmegleichung lautet

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{rcll}{\frac {\partial u}{\partial t}}(x,t)&=&\Delta _{x}u(x,t)&{\text{in }}\mathbb {R} ^{n}\times ]0,\infty [\\u(x,t)&=&g(x,t)&{\text{auf }}\mathbb {R} ^{n}\times \{t=0\}\,.\end{array}}\right.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d3a0f2fd31a5741ceaa14c63ba9945fab32c3fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.516ex; height:7.509ex;" alt="{\displaystyle \left\{{\begin{array}{rcll}{\frac {\partial u}{\partial t}}(x,t)&=&\Delta _{x}u(x,t)&{\text{in }}\mathbb {R} ^{n}\times ]0,\infty [\\u(x,t)&=&g(x,t)&{\text{auf }}\mathbb {R} ^{n}\times \{t=0\}\,.\end{array}}\right.}" loading="lazy"></dd></dl>

Hierbei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{x}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{x}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bbf9f91a144687ca2a31307ad731850f8ee44aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.108ex; height:2.509ex;" alt="{\displaystyle \Delta _{x}}" loading="lazy"> den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a>, der nur auf die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-Variablen wirkt. Anwenden der Fourier-Transformation auf beide Gleichungen bezüglich der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-Variablen und Anwenden der Differentiationsregel ergibt

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{rcll}{\mathcal {F}}\left({\frac {\partial u}{\partial t}}\right)(\xi ,t)&=&-|\xi |^{2}{\mathcal {F}}(u)(\xi ,t)&{\text{in }}\mathbb {R} ^{n}\times ]0,\infty [\\{\mathcal {F}}(u)(\xi ,t)&=&{\mathcal {F}}(g)(\xi ,t)&{\text{auf }}\mathbb {R} ^{n}\times \{t=0\}\,.\end{array}}\right.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29ea96b1150558d2632c19734efefc6bae5cbb74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:57.207ex; height:8.176ex;" alt="{\displaystyle \left\{{\begin{array}{rcll}{\mathcal {F}}\left({\frac {\partial u}{\partial t}}\right)(\xi ,t)&=&-|\xi |^{2}{\mathcal {F}}(u)(\xi ,t)&{\text{in }}\mathbb {R} ^{n}\times ]0,\infty [\\{\mathcal {F}}(u)(\xi ,t)&=&{\mathcal {F}}(g)(\xi ,t)&{\text{auf }}\mathbb {R} ^{n}\times \{t=0\}\,.\end{array}}\right.}" loading="lazy"></dd></dl>

Hierbei handelt es sich nun um eine <a href="Gew%C3%B6hnliche_Differentialgleichung" title="Gewöhnliche Differentialgleichung">gewöhnliche Differentialgleichung</a>, die die Lösung

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(u)(\xi ,t)=\mathrm {e} ^{-t|\xi |^{2}}{\mathcal {F}}(g)(\xi ,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/500fd80bcf2468d1efe1f3e5e25904f5e0bd4368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.053ex; height:3.676ex;" alt="{\displaystyle {\mathcal {F}}(u)(\xi ,t)=\mathrm {e} ^{-t|\xi |^{2}}{\mathcal {F}}(g)(\xi ,t)}" loading="lazy"></dd></dl>

hat. Daraus folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle u(x,t)={\mathcal {F}}^{-1}\left(\exp(-t|\xi |^{2}){\mathcal {F}}(g)\right)(x,t)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle u(x,t)={\mathcal {F}}^{-1}\left(\exp(-t|\xi |^{2}){\mathcal {F}}(g)\right)(x,t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9f61ca6af89f8ca8022e946eb8c5881243744b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.578ex; height:4.843ex;" alt="{\displaystyle \textstyle u(x,t)={\mathcal {F}}^{-1}\left(\exp(-t|\xi |^{2}){\mathcal {F}}(g)\right)(x,t)}" loading="lazy"> und aufgrund des Faltungstheorems gilt

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<annotation encoding="application/x-tex">{\displaystyle u(x,t)={\frac {g(x,t)*F(x,t)}{(2\pi )^{\frac {n}{2}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a1a70a6e306e701d32cf7916baeb255ce9e051b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.353ex; height:7.343ex;" alt="{\displaystyle u(x,t)={\frac {g(x,t)*F(x,t)}{(2\pi )^{\frac {n}{2}}}}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(F)(\xi ,t)=\exp(-t|\xi |^{2}).}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(F)(\xi ,t)=\exp(-t|\xi |^{2}).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21a4bf4d10bb928ab4f2272b520f8631ae00498f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.322ex; height:3.343ex;" alt="{\displaystyle {\mathcal {F}}(F)(\xi ,t)=\exp(-t|\xi |^{2}).}" loading="lazy"> Daraus folgt

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<annotation encoding="application/x-tex">{\displaystyle F(x,t)={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} x\cdot y-t|y|^{2}}\mathrm {d} y={\frac {1}{(2t)^{\frac {n}{2}}}}\mathrm {e} ^{\frac {-|x|^{2}}{4t}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8734f2fd1bf557d9e028df821630cb4220c21fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:48.226ex; height:7.509ex;" alt="{\displaystyle F(x,t)={\frac {1}{(2\pi )^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} x\cdot y-t|y|^{2}}\mathrm {d} y={\frac {1}{(2t)^{\frac {n}{2}}}}\mathrm {e} ^{\frac {-|x|^{2}}{4t}}\,.}" loading="lazy"></dd></dl>

Das ist die <a href="Fundamentall%C3%B6sung" title="Fundamentallösung">Fundamentallösung</a> der Wärmegleichung. Die Lösung des hier betrachteten Anfangswertproblems hat daher die Darstellung

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<annotation encoding="application/x-tex">{\displaystyle u(x,t)={\frac {g(x,t)*F(x,t)}{(2\pi )^{\frac {n}{2}}}}={\frac {1}{(4\pi t)^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{-{\frac {|x-\xi |^{2}}{4t}}}g(\xi )\mathrm {d} \xi \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e930a1ee32fd0f89aafd89dad181ff015d6af7ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:55.378ex; height:7.509ex;" alt="{\displaystyle u(x,t)={\frac {g(x,t)*F(x,t)}{(2\pi )^{\frac {n}{2}}}}={\frac {1}{(4\pi t)^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{-{\frac {|x-\xi |^{2}}{4t}}}g(\xi )\mathrm {d} \xi \,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Tabelle_wichtiger_Fourier-Transformations-Paare">Tabelle wichtiger Fourier-Transformations-Paare</h2></div>

In diesem Kapitel folgt eine Zusammenstellung wichtiger Fourier-Transformations-Paare.

<th>Signal</th>

<th>Fouriertransformierte Kreisfrequenz</th>

<th>Fouriertransformierte Frequenz</th>

<th>Hinweise

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<annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d5b376bac4f3f231eb1d80a3b82fde03eb4a007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.181ex; height:6.343ex;" alt="{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533ec909871e75c727cf7add4095bd9b9c4c586b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.006ex; height:6.009ex;" alt="{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{-\mathrm {i} a\omega }G(\omega )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b07d022e7212060c5bb1444976a8119f0e7958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:3.176ex;" alt="{\displaystyle \mathrm {e} ^{-\mathrm {i} a\omega }G(\omega )}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{-\mathrm {i} 2\pi af}G(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19587ff713716f15b5864ea07a4b3e088b332eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.453ex; height:3.176ex;" alt="{\displaystyle \mathrm {e} ^{-\mathrm {i} 2\pi af}G(f)}" loading="lazy">

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<td>Zeitverschiebung

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} at}g(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0856880cb0dba9cffcdf562d3d5e20f3a873ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.95ex; height:3.176ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} at}g(t)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle G(\omega -a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dc45d95db163190ea40ce0cfc3d576c4ec43e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.152ex; height:2.843ex;" alt="{\displaystyle G(\omega -a)}" loading="lazy">

</td>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\left(f-{\frac {a}{2\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3722eff0568312d0a0f8b2be750e653c2358845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.439ex; height:4.843ex;" alt="{\displaystyle G\left(f-{\frac {a}{2\pi }}\right)}" loading="lazy">

</td>

<td>Frequenzverschiebung

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(at)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f71a94e6c33229c33041b9efe49fc324ed35cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.995ex; height:2.843ex;" alt="{\displaystyle g(at)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca3d5c81f243bbe123dba5880f5f70d31242d2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.631ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92f8c793d23178fe1e50c95e0648c35b7a5e0975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.109ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)}" loading="lazy">

</td>

<td>Frequenzskalierung

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{(n)}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c7c8e1e6342e37f5a7d0da5e6fc17b48397a985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.265ex; height:3.343ex;" alt="{\displaystyle g^{(n)}(t)}" loading="lazy">

</td>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathrm {i} \omega )^{n}G(\omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43a3e2a64525e462ed4dbf43cc4d7a38eb6df1de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.202ex; height:2.843ex;" alt="{\displaystyle (\mathrm {i} \omega )^{n}G(\omega )}" loading="lazy">

</td>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathrm {i} 2\pi f)^{n}G(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf71e474b6e433cbc12e5a3defa4ace2a212a8b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.362ex; height:2.843ex;" alt="{\displaystyle (\mathrm {i} 2\pi f)^{n}G(f)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> eine <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> und g eine <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Funktion</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{(n)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{(n)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb043ee1364ddf59182639f425acc95d82110672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:3.176ex;" alt="{\displaystyle g^{(n)}}" loading="lazy"> bezeichnet die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-te Ableitung von g.

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Quadratisch_integrierbare_Funktionen">Quadratisch integrierbare Funktionen</h3></div>

<th>Signal</th>

<th>Fouriertransformierte Kreisfrequenz</th>

<th>Fouriertransformierte Frequenz</th>

<th>Hinweise

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b84f700860ee7af27797d11ddfad3d185eb7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}" loading="lazy">

</td>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d5b376bac4f3f231eb1d80a3b82fde03eb4a007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.181ex; height:6.343ex;" alt="{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}" loading="lazy">

</td>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533ec909871e75c727cf7add4095bd9b9c4c586b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.006ex; height:6.009ex;" alt="{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(-{\frac {at^{2}}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01149692d93aad9649d67bd2ce21f74b10a8513b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.742ex; height:6.343ex;" alt="{\displaystyle \exp \left(-{\frac {at^{2}}{2}}\right)}" loading="lazy">

</td>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {a}}}\cdot \exp \left(-{\frac {\omega ^{2}}{2a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee15143c619a0e4a64da0f8bf499343086abbebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.799ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {a}}}\cdot \exp \left(-{\frac {\omega ^{2}}{2a}}\right)}" loading="lazy">

</td>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2\pi }{a}}}\exp \left(-{\frac {2\pi }{a}}\cdot \pi f^{2}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/278530be43da0dd7014448efaa36c31daa86b250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.54ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {2\pi }{a}}}\exp \left(-{\frac {2\pi }{a}}\cdot \pi f^{2}\right)}" loading="lazy">

</td>

<td>Die Gaußsche Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(-t^{2}/2)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(-t^{2}/2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f93e220c2fdee9eb8babf4398622648dedacb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.389ex; height:3.176ex;" alt="{\displaystyle \exp(-t^{2}/2)}" loading="lazy"> ergibt fouriertransformiert wieder dieselbe Funktion. Für die Integrierbarkeit muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} (a)>0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} (a)>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87c5b07835b919c881c0365b3ffc5f158a088c54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.043ex; height:2.843ex;" alt="{\displaystyle \mathrm {Re} (a)>0}" loading="lazy"> sein.

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} (at)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3901e0ef4eeaf0b1c28456cd9a69edc2083995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.76ex; height:2.843ex;" alt="{\displaystyle \operatorname {rect} (at)}" loading="lazy">

</td>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt {2\pi }}|a|}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/012a702b4462fbd1abed9dc702ccb52c96081d95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.693ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{{\sqrt {2\pi }}|a|}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {f}{a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7fe52a690224687b8103edeaf24a7dc780e6b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.463ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {f}{a}}\right)}" loading="lazy">

</td>

<td>Die <a href="Rechteckfunktion" title="Rechteckfunktion">Rechteckfunktion</a> und die <a href="Sinc-Funktion" title="Sinc-Funktion">sinc-Funktion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31b4ad779f660c913cfdf6c0f13591fd4a1c12f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.086ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}" loading="lazy">).

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (at)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20f4b123cfded6f7c5e40e199c831f664611a18a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.767ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (at)}" loading="lazy">

</td>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt {2\pi }}|a|}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b710a25ce874aff2de3e5a9dec74ea4d0b91282a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.686ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{{\sqrt {2\pi }}|a|}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {f}{a}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea5f0521b1519f973a0727ae532fdda13a1d5252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.456ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {f}{a}}\right)}" loading="lazy">

</td>

<td>Die Rechteckfunktion ist ein idealisierter Tiefpassfilter, und die sinc-Funktion ist die akausale Stoßantwort eines solchen Filters (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31b4ad779f660c913cfdf6c0f13591fd4a1c12f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.086ex; height:2.843ex;" alt="{\displaystyle \operatorname {sinc} (x)=\sin(\pi x)/(\pi x)}" loading="lazy">).

</td></tr>

<tr>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(-a|t|\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9403fffc0237125fa4d49cc6daad3b0106a30a7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.533ex; height:2.843ex;" alt="{\displaystyle \exp \left(-a|t|\right)}" loading="lazy">

</td>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {a}{\omega ^{2}+a^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/532325bfaeb583fb33564c7a42a534443c5ad4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.953ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {a}{\omega ^{2}+a^{2}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2a}{(2\pi f)^{2}+a^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f102b0fbdc83c8430ec9f19a3cfff6570bd32498.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.597ex; height:6.009ex;" alt="{\displaystyle {\frac {2a}{(2\pi f)^{2}+a^{2}}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a>0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47586c689085690d15621968c72a61073b32357e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.138ex; height:2.176ex;" alt="{\displaystyle a>0.}" loading="lazy"> Die FT der um den Ursprung <a href="Exponentialfunktion" title="Exponentialfunktion">exponentiell</a> abfallenden Funktion ist eine <a href="Lorentzkurve" title="Lorentzkurve">Lorentzkurve</a>.

</td></tr>

<tr>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t^{2}+a^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/886164fe055619e29463a6d0df6803f4e6b8f476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.854ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{t^{2}+a^{2}}}}" loading="lazy">

</td>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{a}}\exp \left(-a|\omega |\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaba0bec017ce1e5fa69e29637c2db0e25587f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.084ex; height:6.343ex;" alt="{\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{a}}\exp \left(-a|\omega |\right)}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{a}}\exp \left(-2\pi a|f|\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddf6c5b231fed0aa54868058d9560e98c10c3f3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.022ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{a}}\exp \left(-2\pi a|f|\right)}" loading="lazy">

</td>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Distributionen">Distributionen</h3></div>

<th>Signal</th>

<th>Fouriertransformierte Kreisfrequenz</th>

<th>Fouriertransformierte Frequenz</th>

<th>Hinweise

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b84f700860ee7af27797d11ddfad3d185eb7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}" loading="lazy">

</td>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d5b376bac4f3f231eb1d80a3b82fde03eb4a007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.181ex; height:6.343ex;" alt="{\displaystyle G(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} \omega t}\mathrm {d} t}" loading="lazy">

</td>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533ec909871e75c727cf7add4095bd9b9c4c586b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.006ex; height:6.009ex;" alt="{\displaystyle G(f)=\int _{-\infty }^{\infty }g(t)\mathrm {e} ^{-\mathrm {i} 2\pi ft}\mathrm {d} t}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\mathrm {i} at}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaa6c661ed4923fe190580adfa5ada61586b1fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.185ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{\mathrm {i} at}}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83a285410b0a5cb1a716b3a5ebdefba3abdaeca6.svg" 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)}" loading="lazy">

</td>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta \left(f-{\frac {a}{2\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ace1d2e5508dd49fa020d28685afebbd8df10eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.661ex; height:4.843ex;" alt="{\displaystyle \delta \left(f-{\frac {a}{2\pi }}\right)}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(at)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/284bfbe35ad1c6a69786f8fa430204c2c82fd1b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.99ex; height:2.843ex;" alt="{\displaystyle \cos(at)}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d3054f23623d1c05258b7bea1d5785995ba61db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.855ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta (f-{\frac {a}{2\pi }})+\delta (f+{\frac {a}{2\pi }})}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e6caefc61cd9460d458f8f78573e53c17eaba6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.83ex; height:6.009ex;" alt="{\displaystyle {\frac {\delta (f-{\frac {a}{2\pi }})+\delta (f+{\frac {a}{2\pi }})}{2}}}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(at)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9281a0c76cc83a3a6a288e568d9b94e4a41e1a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.734ex; height:2.843ex;" alt="{\displaystyle \sin(at)}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)-\delta (\omega +a)}{2\mathrm {i} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b16b2f14e6b36b29642192a587264e46f00baf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.855ex; height:5.676ex;" alt="{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)-\delta (\omega +a)}{2\mathrm {i} }}}" loading="lazy">

</td>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta (f-{\frac {a}{2\pi }})-\delta (f+{\frac {a}{2\pi }})}{2\mathrm {i} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e37084d1f7f7126c154cde4beff191ecd139c6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.83ex; height:6.009ex;" alt="{\displaystyle {\frac {\delta (f-{\frac {a}{2\pi }})-\delta (f+{\frac {a}{2\pi }})}{2\mathrm {i} }}}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0abb8feafada8bb80cabe3ba32346cbaff464f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t^{n}}" loading="lazy">

</td>

<msup>

</mrow>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} ^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b60b152f5e4e3487adb222a7e8b7a4e339e5259c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.102ex; height:3.343ex;" alt="{\displaystyle \mathrm {i} ^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )}" loading="lazy">

</td>

<msup>

<mrow>

<mfrac>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\mathrm {i} }{2\pi }}\right)^{n}\delta ^{(n)}(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fee8b85bb2899d09500922a8a2ed67966c3d9ee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.61ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\mathrm {i} }{2\pi }}\right)^{n}\delta ^{(n)}(f)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> eine <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{(n)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta ^{(n)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6af7d40340694e369f1cd5d1a316fa3f7f1ebea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.843ex;" alt="{\displaystyle \delta ^{(n)}}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-te Ableitung der <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a>.

</td></tr>

<tr>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t^{n}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44ce9b8919b2a6629846315ced6cae7c70e06244.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.894ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{t^{n}}}}" loading="lazy">

</td>

</mrow>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\mathrm {i} {\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-\mathrm {i} \omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d9db5aed44f6bb36f8fbca5c9cb18e2244dce4.svg" 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 -\mathrm {i} {\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-\mathrm {i} \omega )^{n-1}}{(n-1)!}}\operatorname {sgn} (\omega )}" loading="lazy">

</td>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\mathrm {i} \pi {\frac {(-\mathrm {i} 2\pi f)^{n-1}}{(n-1)!}}\operatorname {sgn}(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fed47e7ddd20b73f719ed92b650a9ee575aaf6ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.826ex; height:6.676ex;" alt="{\displaystyle -\mathrm {i} \pi {\frac {(-\mathrm {i} 2\pi f)^{n-1}}{(n-1)!}}\operatorname {sgn} (f)}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b031d2f26b3595705817ff565fa8e7520eb816a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.02ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} (t)}" loading="lazy">

</td>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{\mathrm {i} \ \omega }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5fea235d6a965f3e8f0ce727edda00ac0444d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.68ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{\mathrm {i} \ \omega }}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\mathrm {i} \pi f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a52998feddc47782a09505519b945c7a2d9ae2fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.094ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\mathrm {i} \pi f}}}" loading="lazy">

</td>

<td>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/415d180b65fdc75e95cb50e5786e6c56989dd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.457ex; height:2.843ex;" alt="{\displaystyle \Theta (t)}" loading="lazy">

</td>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{\mathrm {i} \pi \omega }}+\delta (\omega )\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/275d7d47c9148f32e9d762f33b03baddce5ebdb4.svg" 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}{\mathrm {i} \pi \omega }}+\delta (\omega )\right)}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left({\frac {1}{\mathrm {i} \pi f}}+\delta (f)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb1315444d8cd82af1a679cfc09ec8717feee360.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.877ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2}}\left({\frac {1}{\mathrm {i} \pi f}}+\delta (f)\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta (t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/415d180b65fdc75e95cb50e5786e6c56989dd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.457ex; height:2.843ex;" alt="{\displaystyle \Theta (t)}" loading="lazy"> ist der Einheitssprung (<a href="Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a>).

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a13edb5e0e4fec83929d6bb3b50fcd5eac5b36d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.142ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

<mrow>

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</mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\frac {2\pi }{T}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa91eec43e34b1c5fe2d8865c6b649edbb68fb73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.846ex; height:7.009ex;" alt="{\displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\frac {2\pi }{T}}\right)}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfde496fa3fc90f38f2c304800728e24c569b868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.752ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)}" loading="lazy">

</td>

<td>Das Signal heißt <a href="Dirac-Kamm" title="Dirac-Kamm">Dirac-Kamm</a>.

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">Diskrete Fourier-Transformation</a></li>

<li><a href="Kurzzeit-Fourier-Transformation" title="Kurzzeit-Fourier-Transformation">Kurzzeit-Fourier-Transformation</a></li>

<li><a href="Fouriertransformation_f%C3%BCr_zeitdiskrete_Signale" title="Fouriertransformation für zeitdiskrete Signale">Fouriertransformation für zeitdiskrete Signale</a></li>

<li><a href="Schnelle_Fourier-Transformation" title="Schnelle Fourier-Transformation">Schnelle Fourier-Transformation</a></li>

<li><a href="IFFT" title="IFFT">Inverse schnelle Fourier-Transformation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Rolf Brigola: <cite style="font-style:italic">Fourier-Analysis und Distributionen</cite>. edition swk, Hamburg 2013, ISBN 978-3-8495-2892-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fourier-Transformation&rft.au=Rolf+Brigola&rft.btitle=Fourier-Analysis+und+Distributionen&rft.date=2013&rft.genre=book&rft.isbn=9783849528928&rft.place=Hamburg&rft.pub=edition+swk" style="display:none"> </li>

<li><a href="Salomon_Bochner" title="Salomon Bochner">S. Bochner</a>, <a href="K._Chandrasekharan" title="K. Chandrasekharan">K. Chandrasekharan</a>: Fourier Transforms. Princeton University Press, Princeton NJ 1949 (Annals of mathematics studies 19, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220066-2313%22&key=cql">0066-2313</a>).</li>

<li><a href="Otto_F%C3%B6llinger" title="Otto Föllinger">Otto Föllinger</a>: Laplace-, Fourier- und z-Transformation. Bearbeitet von Mathias Kluwe. 8. überarbeitete Auflage. Hüthig, Heidelberg 2003, ISBN 3-7785-2911-0 (Studium).</li>

<li><a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Lars Hörmander</a>: The Analysis of Linear Partial Differential Operators I. Second Edition. Springer-Verlag, ISBN 3-540-52345-6.</li>

<li>Burkhard Lenze: Einführung in die Fourier-Analysis. 3. durchgesehene Auflage. Logos Verlag, Berlin 2010, ISBN 3-931216-46-2.</li>

<li><a href="Michael_James_Lighthill" title="Michael James Lighthill">M. J. Lighthill</a>: Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge 2003, ISBN 0-521-09128-4 (Cambridge Monographs on Mechanics and Applied Mathematics).</li>

<li>P. I. Lizorkin: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Fourier Transform</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/fourier_transform">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Fourier-Transformation&rft.atitle=Fourier+Transform&rft.au=P.+I.+Lizorkin&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </li>

<li>Athanasios Papoulis: The Fourier Integral and Its Applications. Reissued. McGraw-Hill, New York NY u. a. 1987, ISBN 0-07-048447-3 (McGraw-Hill Classic Textbook Reissue Series).</li>

<li>Lothar Papula: Mathematische Formelsammlung. 11. Auflage. Springer Verlag. Wiesbaden 2014, ISBN 978-3-8348-2311-3.</li>

<li>Herbert Sager: Fourier-Transformation. 1. Auflage. vdf Hochschulverlag AG an der ETH Zürich, Zürich 2012, ISBN 978-3-7281-3393-9.</li>

<li><a href="Elias_M._Stein" class="mw-redirect" title="Elias M. Stein">Elias M. Stein</a>, Rami Shakarchi: Princeton Lectures in Analysis. Band 1: Fourier Analysis. An Introduction. Princeton University Press, Princeton NJ 2003, ISBN 0-691-11384-X.</li>

<li><a href="Dirk_Werner_(Mathematiker)" title="Dirk Werner (Mathematiker)">Dirk Werner</a>: Funktionalanalysis. Springer-Verlag, 6. Auflage, ISBN 978-3-540-72533-6.</li>

<li>Jörg Lange, <a href="Tatjana_Lange" title="Tatjana Lange">Tatjana Lange</a>: Fourier-Transformation zur Signal- und Systembeschreibung. Kompakt, visuell, intuitiv verständlich. Springer Vieweg, 2019, ISBN 978-3-658-24849-9.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fourier_transformation?uselang=de">Commons: Fourier transformation</a> – Sammlung von Bildern, Videos und Audiodateien</div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FourierTransform.html">Fourier Transform.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=spUNpyF58BY">Was ist eine Fourier-Transformation? Eine visuelle Einführung</a> auf <a href="YouTube" title="YouTube">YouTube</a>, abgerufen am 15. April 2023.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Martin Donner: <a rel="nofollow" class="external text" href="https://www.musikundmedien.hu-berlin.de/de/medienwissenschaft/medientheorien/hausarbeiten_essays/pdfs/fourier-neue-medien-web.pdf">Fouriers Beitrag zur Geschichte der Neuen Medien.</a> In: Humboldt-Universität zu Berlin. 2006, abgerufen am 30. Juli 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFourier-Transformation&rft.title=Fouriers+Beitrag+zur+Geschichte+der+Neuen+Medien&rft.description=Fouriers+Beitrag+zur+Geschichte+der+Neuen+Medien&rft.identifier=https%3A%2F%2Fwww.musikundmedien.hu-berlin.de%2Fde%2Fmedienwissenschaft%2Fmedientheorien%2Fhausarbeiten_essays%2Fpdfs%2Ffourier-neue-medien-web.pdf&rft.creator=Martin+Donner&rft.date=2006&rft.language=de">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Dirk Schulze: <a rel="nofollow" class="external text" href="https://tu-dresden.de/ing/informatik/ti/vlsi/ressourcen/dateien/dateien_studium/dateien_lehstuhlseminar/vortraege_lehrstuhlseminar/hs_ss08/Vortrag.pdf?lang=en">Digitale Audiokodierung mit MP3, Varianten und Anwendungsgebiete.</a> In: Technische Universität Dresden. 2008, abgerufen am 30. Juli 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFourier-Transformation&rft.title=Digitale+Audiokodierung+mit++MP3%2C+Varianten+und++Anwendungsgebiete&rft.description=Digitale+Audiokodierung+mit++MP3%2C+Varianten+und++Anwendungsgebiete&rft.identifier=https%3A%2F%2Ftu-dresden.de%2Fing%2Finformatik%2Fti%2Fvlsi%2Fressourcen%2Fdateien%2Fdateien_studium%2Fdateien_lehstuhlseminar%2Fvortraege_lehrstuhlseminar%2Fhs_ss08%2FVortrag.pdf%3Flang%3Den&rft.creator=Dirk+Schulze&rft.date=2008">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Johannes Klotz: <a rel="nofollow" class="external text" href="https://diglib.uibk.ac.at/ulbtirolhs/content/titleinfo/3580370/full.pdf">Grundlagen der Fourier-Transformation und deren Anwendung in der Magnetresonanztomographie (MRT).</a> Universität Innsbruck, 30. April 2019, abgerufen am 30. Juli 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFourier-Transformation&rft.title=Grundlagen+der+Fourier-Transformation+und+deren+Anwendung+in+der+Magnetresonanztomographie+%28MRT%29&rft.description=Grundlagen+der+Fourier-Transformation+und+deren+Anwendung+in+der+Magnetresonanztomographie+%28MRT%29&rft.identifier=https%3A%2F%2Fdiglib.uibk.ac.at%2Fulbtirolhs%2Fcontent%2Ftitleinfo%2F3580370%2Ffull.pdf&rft.creator=Johannes+Klotz&rft.publisher=Universit%C3%A4t+Innsbruck&rft.date=2019-04-30">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Beweis mittels Einsetzen der inversen Fouriertransformierten, z. B. wie in Tilman Butz: Fouriertransformation für Fußgänger. Ausgabe 7, Springer DE, 2011, ISBN 978-3-8348-8295-0, S. 53, <a rel="nofollow" class="external text" href="http://books.google.de/books?id=RNsgEcp_a4MC&lpg=PA51&ots=meOb8yOqO-&dq=faltung%20von%20fouriertransformierter&hl=de&pg=PA53#v=onepage&q=faltung%20von%20fouriertransformierter&f=false">Google Books.</a>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Helmut Fischer, Helmut Kaul: Mathematik für Physiker. Band 2: Gewöhnliche und partielle Differentialgleichungen, mathematische Grundlagen der Quantenmechanik. 2. Auflage. B.G. Teubner, Wiesbaden 2004, ISBN 3-519-12080-1, § 12, Abschn. 4.2, S. 300–301.

</li>

</ol>

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Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4018014-1">4018014-1</a> </div>

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