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<title>Temperierte Distribution</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Temperierte Distribution</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>temperierte Distribution</b> ist ein Objekt aus der <a href="Distributionentheorie" class="mw-redirect" title="Distributionentheorie">Distributionentheorie</a>, einem mathematischen Teilgebiet der <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>. Eine temperierte Distribution ist ein Spezialfall einer <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distribution</a>. <a href="Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a> führte 1947 den Raum der temperierten Distributionen ein, um die <a href="Kontinuierliche_Fourier-Transformation" class="mw-redirect" title="Kontinuierliche Fourier-Transformation">Fourier-Transformation</a> in seine Distributionentheorie integrieren zu können.

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

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a></i></div>

<p>Um temperierte Distributionen definieren zu können, wird zuerst der Raum der <i>schnell fallenden Funktionen</i> erläutert. Schnell fallende Funktionen sind unendlich oft differenzierbar und streben im Unendlichen so schnell gegen null, dass sie und alle ihre Ableitungen schneller als jede <a href="Polynomfunktion" class="mw-redirect" title="Polynomfunktion">Polynomfunktion</a> fallen. Die Menge all dieser Funktionen wird auch als <i>Schwartz-Raum</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0078d18e4675b6e7e2acb6c2c25c65294193e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"></span> bezeichnet und ist durch

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})=\{\phi \in C^{\infty }(\mathbb {R} ^{n})\,|\,\forall \alpha \in \mathbb {N} _{0}^{n},\beta \in \mathbb {N} _{0}^{n}\;\exists C\geq 0:\;\sup _{x\in \mathbb {R} ^{n}}|x^{\alpha }D^{\beta }\phi (x)|\leq C\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c963cd7c1de4aae3c290c32175b8e1c1673b9c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:73.786ex; height:5.009ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})=\{\phi \in C^{\infty }(\mathbb {R} ^{n})\,|\,\forall \alpha \in \mathbb {N} _{0}^{n},\beta \in \mathbb {N} _{0}^{n}\;\exists C\geq 0:\;\sup _{x\in \mathbb {R} ^{n}}|x^{\alpha }D^{\beta }\phi (x)|\leq C\}}" loading="lazy"></span></dd></dl>

<p>definiert. Durch die <a href="Halbnorm" title="Halbnorm">Halbnormen</a>

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<annotation encoding="application/x-tex">{\displaystyle \|f\|_{N}=\sup _{x\in \mathbb {R} ^{n}}\max _{|\alpha |,\,|\beta |<N}|x^{\alpha }D^{\beta }f(x)|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3874228c4d4728568c1b971927fefad1dd41502c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.185ex; height:5.009ex;" alt="{\displaystyle \|f\|_{N}=\sup _{x\in \mathbb {R} ^{n}}\max _{|\alpha |,\,|\beta |<N}|x^{\alpha }D^{\beta }f(x)|}" loading="lazy"></span></dd></dl>

<p>wird der Schwartz-Raum zu einem <a href="Metrisierbarer_lokalkonvexer_Raum" title="Metrisierbarer lokalkonvexer Raum">metrisierbaren lokalkonvexen Raum</a>. Die Besonderheit dieses Raumes ist, dass die <a href="Kontinuierliche_Fourier-Transformation" class="mw-redirect" title="Kontinuierliche Fourier-Transformation">Fourier-Transformation</a> ein <a href="Automorphismus" title="Automorphismus">Automorphismus</a> auf diesem ist. Außerdem ist der Raum in allen <a href="Sobolew-Raum" class="mw-redirect" title="Sobolew-Raum">Sobolew-Räumen</a> enthalten. Der Raum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}(\mathbb {R} ^{n})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40967820fe4c745ac9e5246e47e5897b08b2dc9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.498ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(\mathbb {R} ^{n})}" loading="lazy"></span> der <a href="Testfunktion" title="Testfunktion">Testfunktionen</a> lässt sich <a href="Stetige_Abbildung" class="mw-redirect" title="Stetige Abbildung">stetig</a> in den Schwartz-Raum <a href="Einbettung_(Mathematik)" title="Einbettung (Mathematik)">einbetten</a> und liegt in diesem <a href="Dichte_Teilmenge" title="Dichte Teilmenge">dicht</a>.

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

<p>Eine temperierte Distribution ist ein stetiges, <a href="Lineare_Abbildung" title="Lineare Abbildung">lineares</a> <a href="Funktional" title="Funktional">Funktional</a> auf dem Schwartz-Raum, also eine stetige lineare Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})\to \mathbb {C} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccc7862ff51318a2cdcb9d750b50d1d0d942788c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.49ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})\to \mathbb {C} }" loading="lazy"></span>. Da die Menge der temperierten Distributionen der Definition nach den topologischen <a href="Dualraum" title="Dualraum">Dualraum</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0078d18e4675b6e7e2acb6c2c25c65294193e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"></span> bildet, wird dieser Raum mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78ca4a0bc72d2677618a39482625e3f749990455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span> notiert. Aufgrund dieser Dualität spricht man auch von den langsam wachsenden Distributionen im Gegensatz zu den schnell fallenden Funktionen.

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

<ul><li>Die Klasse der Distributionen mit kompaktem Träger ist eine echte Untermenge des Raums der temperierten Distributionen. Ein Beispiel einer Distribution mit kompaktem Träger ist die <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a>.</li>

<li><a href="Dirac-Kamm" title="Dirac-Kamm">Dirac-Kamm</a></li>

<li>Alle Distributionen, die durch eine <a href="Polynomfunktion" class="mw-redirect" title="Polynomfunktion">Polynomfunktion</a> erzeugt werden, sind temperierte Distributionen. Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} )\ni \phi \mapsto \int _{\mathbb {R} }P(x)\phi (x)\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fef09b0e2bc6eafe75e5a2121f72ddd6ea474c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.949ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} )\ni \phi \mapsto \int _{\mathbb {R} }P(x)\phi (x)\mathrm {d} x}" loading="lazy"></span></dd></dl></dd>

<dd>eine temperierte Distribution. Diese Distributionen sind im Gegensatz zur Delta-Distribution beziehungsweise zum Dirac-Kamm <a href="Regul%C3%A4re_Distribution" class="mw-redirect" title="Reguläre Distribution">reguläre Distributionen</a>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Gelfandsches_Raumtripel">Gelfandsches Raumtripel</h2></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Gelfand-Tripel" title="Gelfand-Tripel">Gelfand-Tripel</a></i></div>

<p>Der Schwartz-Raum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0078d18e4675b6e7e2acb6c2c25c65294193e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"></span> liegt dicht im <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} ^{n})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f2e538f9f83f3aeb3d48b399fcdb29918af33f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.405ex; height:3.176ex;" alt="{\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} ^{n})}" loading="lazy"></span> der <a href="Quadratintegrierbar" class="mw-redirect" title="Quadratintegrierbar">quadratintegrierbaren Funktionen</a>. Aus diesem Grund gilt für ihre Dualräume die Inklusion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L^{2}(\mathbb {R} ^{n}))'\subset {\mathcal {S}}'(\mathbb {R} ^{n})}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4c48a7e88bd3c03e4e0cb047857e0ce2b28e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.843ex; height:3.176ex;" alt="{\displaystyle (L^{2}(\mathbb {R} ^{n}))'\subset {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span> und aus dem <a href="Satz_von_Riesz-Fischer" class="mw-redirect" title="Satz von Riesz-Fischer">Satz von Riesz-Fischer</a> folgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} ^{n})\cong (L^{2}(\mathbb {R} ^{n}))'.}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d0dee42671907e9f10c3320865545623663200b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.925ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{n})\cong (L^{2}(\mathbb {R} ^{n}))'.}" loading="lazy"></span> Dies führt insgesamt zu der Inklusion

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})\subset L^{2}(\mathbb {R} ^{n})\hookrightarrow {\mathcal {S}}'(\mathbb {R} ^{n}).}</annotation>

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<p>Die stetige <a href="Einbettung_(Mathematik)" title="Einbettung (Mathematik)">Einbettung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon L^{2}(\mathbb {R} ^{n})\hookrightarrow {\mathcal {S}}'(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle i\colon L^{2}(\mathbb {R} ^{n})\hookrightarrow {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f4e097c745663c57790c0472faaf706d5479356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.994ex; height:3.176ex;" alt="{\displaystyle i\colon L^{2}(\mathbb {R} ^{n})\hookrightarrow {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span> ist die normale Identifizierung einer Funktion mit einer Distribution. Das heißt, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(\mathbb {R} ^{n})\mapsto \left(\phi \in {\mathcal {S}}(\mathbb {R} ^{n})\mapsto \int _{\mathbb {R} ^{n}}f(x)\phi (x)\mathrm {d} x\right)}</annotation>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e72f3bd2b6d9300b279290027280ded95911aec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.384ex; height:3.176ex;" alt="{\displaystyle ({\mathcal {S}}(\mathbb {R} ^{n}),L^{2}(\mathbb {R} ^{n}))}" loading="lazy"></span> ergibt ein Beispiel für einen erweiterten Hilbertraum, beziehungsweise das Tripel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {S}}(\mathbb {R} ^{n}),L^{2}(\mathbb {R} ^{n}),{\mathcal {S}}'(\mathbb {R} ^{n})\right)}">

<mrow>

<mrow>

</mrow>

<msup>

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</msup>

<msup>

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</mrow>

</msup>

<msup>

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</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

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</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4265f56be78c62f2c48dc9f423c7b8ef2c5f95c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.647ex; height:3.343ex;" alt="{\displaystyle \left({\mathcal {S}}(\mathbb {R} ^{n}),L^{2}(\mathbb {R} ^{n}),{\mathcal {S}}'(\mathbb {R} ^{n})\right)}" loading="lazy"></span> ein Beispiel für ein <a href="Gelfandsches_Raumtripel" class="mw-redirect" title="Gelfandsches Raumtripel">gelfandsches Raumtripel</a> (nach <a href="Israel_Gelfand" class="mw-redirect" title="Israel Gelfand">Israel Gelfand</a>). In allen drei Räumen ist die Fourier-Transformation ein <a href="Automorphismus" title="Automorphismus">Automorphismus</a>.

</p><p>Zu den Werten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \sigma _{c}\left(A\right)}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \in \sigma _{c}\left(A\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e5a511ef9d464375d2d851b94e00e749a4976a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.407ex; height:2.843ex;" alt="{\displaystyle \lambda \in \sigma _{c}\left(A\right)}" loading="lazy"></span> im kontinuierlichen Anteil des <a href="Spektrum_(Operatortheorie)" title="Spektrum (Operatortheorie)">Spektrums</a> eines <a href="Linearer_Operator" title="Linearer Operator">Operators</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> existieren, anders als zu den <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerten</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \sigma _{p}\left(A\right)}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \in \sigma _{p}\left(A\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f33123e932aaa77009fceb6735e6eecee783987.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.522ex; height:3.009ex;" alt="{\displaystyle \lambda \in \sigma _{p}\left(A\right)}" loading="lazy"></span> (also den Werten des Punktspektrums), keine Eigenfunktionen in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>. Es können aber Distributionen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in {\mathcal {S}}'}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\in {\mathcal {S}}'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdce21de104d4b1403d094c974c18dc82a5b564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.679ex; height:2.509ex;" alt="{\displaystyle T\in {\mathcal {S}}'}" loading="lazy"></span> existieren, die an deren Stelle die Eigenwertgleichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda T=AT}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a35d94f2120cc49ca56282b68486ae8fcaabeeec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.469ex; height:2.176ex;" alt="{\displaystyle \lambda T=AT}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e577a88611d41be7e2cd935d34000ca976f9d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" loading="lazy"></span> erfüllen. Weitere Einzelheiten finden sich in Band III der unter <a href="#Literatur">Literatur</a> angegebenen Bücher von Gelfand.

In der Anwendung auf die <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> bedeutet das, dass der Raum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e577a88611d41be7e2cd935d34000ca976f9d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle {\mathcal {S}}'}" loading="lazy"></span> beispielsweise „Eigenfunktionen“ des <a href="Ortsoperator" title="Ortsoperator">Orts-</a> oder <a href="Impulsoperator" title="Impulsoperator">Impulsoperators</a> enthält (in der Standard-Darstellung sind dies <a href="Delta-Distribution" title="Delta-Distribution">δ-Funktionen</a> bzw. <a href="Ebene_Welle" title="Ebene Welle">ebene Wellen</a>), die nicht in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} ^{3})}">

<msup>

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</msup>

<msup>

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</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} ^{3})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fea82a03183a3de22485cf18c7c1e78e83848940.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.179ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} ^{3})}" loading="lazy"></span> enthalten sind, weil das Integral über ihr <a href="Betragsquadrat" title="Betragsquadrat">Betragsquadrat</a> divergiert.

</p>

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

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

<msup>

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<msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/002f32e103345e665cf34a609b5dc6d901ae6351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.078ex; height:3.009ex;" alt="{\displaystyle u\in {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span> eine temperierte Distribution, die <a href="Kontinuierliche_Fourier-Transformation#Fourier-Transformation_im_Raum_der_temperierten_Distributionen" class="mw-redirect" title="Kontinuierliche Fourier-Transformation">Fourier-Transformierte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> ist für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in {\mathcal {S}}(\mathbb {R} ^{n})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/722eae1c976cf2f639e2205227405710827b275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.424ex; height:2.843ex;" alt="{\displaystyle \phi \in {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"></span> definiert durch

</p>

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

</semantics>

</math></span><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"></span>.</dd></dl>

<p>In diesem Kontext ist die Fourier-Transformation auf Funktionen durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\mathcal {F}}(\phi )(\xi )=\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} \langle x,\xi \rangle }\phi (x)\mathrm {d} x}">

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<msub>

<msup>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\mathcal {F}}(\phi )(\xi )=\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} \langle x,\xi \rangle }\phi (x)\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58c71819e164127dff8f42e2dc830599dbf79249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.531ex; height:3.343ex;" alt="{\displaystyle \textstyle {\mathcal {F}}(\phi )(\xi )=\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} \langle x,\xi \rangle }\phi (x)\mathrm {d} x}" loading="lazy"></span> definiert. Es gibt auch eine andere Konvention für die Fourier-Transformation mit dem Vorfaktor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}">

<mfrac>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/942b828bafefa46114cf77e6a12b1fffdce0893a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.179ex; height:4.676ex;" alt="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}" loading="lazy"></span>. Diese wird in diesem Artikel aber nicht verwendet.

</p>

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

<p>Man stattet die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}">

<msup>

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</msup>

<msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78ca4a0bc72d2677618a39482625e3f749990455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span> mit der <a href="Schwach-*-Topologie" title="Schwach-*-Topologie">Schwach-*-Topologie</a> aus. Dann ist die Fourier-Transformation eine stetige, <a href="Bijektive_Funktion" title="Bijektive Funktion">bijektive Abbildung</a> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}">

<msup>

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</msup>

<msup>

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</msup>

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78ca4a0bc72d2677618a39482625e3f749990455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.908ex; height:3.009ex;" alt="{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"></span>. Das Fourier-Urbild von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(u)}">

</mrow>

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

</semantics>

</math></span><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"></span> berechnet sich mit der Formel<br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\phi )(-x)={\frac {1}{(2\pi )^{n}}}{\mathcal {F}}({\mathcal {F}}(u))(\phi )(x).}">

<mfrac>

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<msup>

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</mstyle>

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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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4a360034ad53da0797a98803e489d513c4419f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -2.671ex; width:34.71ex; height:6.009ex;" aria-hidden="true" alt="{\displaystyle u(\phi )(-x)={\frac {1}{(2\pi )^{n}}}{\mathcal {F}}({\mathcal {F}}(u))(\phi )(x).}" loading="lazy"></span>

</p>

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

<msup>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/849a407d43559ab637d97da4dbd1423ee791a469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.967ex; height:2.343ex;" alt="{\displaystyle a\in \mathbb {R} ^{n}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{a}\in S'(\mathbb {R} ^{n})}">

<msub>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55cb8fdf8ca42a8d77db0cf41300dcdf5f76a38e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.887ex; height:3.009ex;" alt="{\displaystyle \delta _{a}\in S'(\mathbb {R} ^{n})}" loading="lazy"></span> die <a href="Deltadistribution" class="mw-redirect" title="Deltadistribution">Deltadistribution</a> zum Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>. Für die Fourier-Transformation gilt dann</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\delta _{a})(\phi )=\delta _{a}({\mathcal {F}}(\phi ))={\mathcal {F}}(\phi )(a)=\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} ax}\phi (x)\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/741482fda9f5a23f0de3659fde2bc483bee9c9e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.657ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}(\delta _{a})(\phi )=\delta _{a}({\mathcal {F}}(\phi ))={\mathcal {F}}(\phi )(a)=\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} ax}\phi (x)\mathrm {d} x}" loading="lazy"></span>.</dd></dl></dd>

<dd>Also entspricht <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\delta _{a})}">

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ed1b7b5ff43e362cbc1d11c4972dadc3addd46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.87ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\delta _{a})}" loading="lazy"></span> der von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto e^{-\mathrm {i} ax}}">

<msup>

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</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f89cf6617f0aa89fd42a125074fcc93dbdaa2a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.805ex; height:2.676ex;" alt="{\displaystyle x\mapsto e^{-\mathrm {i} ax}}" loading="lazy"></span> erzeugten Distribution. Im Fall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy"></span> entspricht also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\delta _{0})}">

</mrow>

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\delta _{0})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed2e263348d11cc35c62d4de8fd6898af9816b98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.823ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\delta _{0})}" loading="lazy"></span> der von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> erzeugten Distribution. Verwendet man bei der Fourier-Transformation noch den Vorfaktor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

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</mfrac>

</mstyle>

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/942b828bafefa46114cf77e6a12b1fffdce0893a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.179ex; height:4.676ex;" alt="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}" loading="lazy"></span> dann ist das Ergebnis des Beispiels die Distribution, die von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}">

<mfrac>

<mrow>

<msup>

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</msup>

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</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/942b828bafefa46114cf77e6a12b1fffdce0893a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.179ex; height:4.676ex;" alt="{\displaystyle {\tfrac {1}{(2\pi )^{n/2}}}}" loading="lazy"></span> erzeugt wird.</dd></dl>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle T_{1}(\phi )=\int _{\mathbb {R} ^{n}}1\cdot \phi (x)\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/141485679ff5eb5e770b9db40f111321454c87f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.561ex; height:3.176ex;" alt="{\displaystyle \textstyle T_{1}(\phi )=\int _{\mathbb {R} ^{n}}1\cdot \phi (x)\mathrm {d} x}" loading="lazy"></span> die von der konstanten Eins-Funktion erzeugte Distribution. Der naheliegende Ansatz, den Ausdruck <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(T_{1})(\phi )}">

</mrow>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(T_{1})(\phi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14fb33d7ab73944a2f6f9d2c0709e6c68edd9e25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.342ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(T_{1})(\phi )}" loading="lazy"></span> zu berechnen, scheitert, da er auf ein nicht absolut konvergentes Integral führt. Zum Lösen benötigt man obiges Beispiel und einen kleinen Trick. Es gilt</li></ul>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(T_{1})(\phi )={\mathcal {F}}({\mathcal {F}}(\delta _{0}))(\phi )=\delta _{0}({\mathcal {F}}({\mathcal {F}}(\phi )))=(2\pi )^{n}\delta _{0}(\phi )(-x)=(2\pi )^{n}\phi (0)=(2\pi )^{n}\delta _{0}(\phi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8ed5dd5aca963cfd2671d9b9ca69ed6d57a25a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:86.775ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(T_{1})(\phi )={\mathcal {F}}({\mathcal {F}}(\delta _{0}))(\phi )=\delta _{0}({\mathcal {F}}({\mathcal {F}}(\phi )))=(2\pi )^{n}\delta _{0}(\phi )(-x)=(2\pi )^{n}\phi (0)=(2\pi )^{n}\delta _{0}(\phi )}" loading="lazy"></span></dd></dl></dd>

<dd>und somit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(T_{1})=(2\pi )^{n}\delta _{0}.}">

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(T_{1})=(2\pi )^{n}\delta _{0}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53c88258f5009432f35ee7c35250f96cdb1c29b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.502ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(T_{1})=(2\pi )^{n}\delta _{0}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Fourier-Laplace-Transformation">Fourier-Laplace-Transformation</h3></div>

<p>In diesem Abschnitt wird die Fourier-Transformation nur für Distributionen mit kompaktem Träger betrachtet. Da die Fourier-Transformation in diesem Kontext besondere Eigenschaften hat, nennt man sie dann Fourier-Laplace-Transformation.

Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {E}}'(\mathbb {R} ^{n})}">

<msup>

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</msup>

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</mstyle>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15157409a321bf1970729163c08d156256bbe24e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.897ex; height:3.009ex;" alt="{\displaystyle u\in {\mathcal {E}}'(\mathbb {R} ^{n})}" loading="lazy"></span> also eine <a href="Distribution_mit_kompaktem_Tr%C3%A4ger" class="mw-redirect" title="Distribution mit kompaktem Träger">Distribution mit kompaktem Träger</a>. Dann ist die Laplace-Fourier-Transformation durch

</p>

<mover>

</mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}(\xi ):=u(e^{-\mathrm {i} \langle \cdot ,\xi \rangle })}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70ecd106099f8f3d80357c531567f76900ae1b36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.028ex; height:3.343ex;" alt="{\displaystyle {\hat {u}}(\xi ):=u(e^{-\mathrm {i} \langle \cdot ,\xi \rangle })}" loading="lazy"></span></dd></dl>

<p>definiert. Dies ist <a href="Wohldefiniertheit" title="Wohldefiniertheit">wohldefiniert</a>, denn man kann zeigen, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28b0daee3e5310b67eb2222b45bea6236d002c69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {u}}}" loading="lazy"></span> eine Funktion ist, welche sogar für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in \mathbb {C} }">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc3b0c7cb1d7a9d77dd75c56d57dc1ef30a79a35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.549ex; height:2.509ex;" alt="{\displaystyle \xi \in \mathbb {C} }" loading="lazy"></span> <a href="Analytische_Funktion" title="Analytische Funktion">analytisch</a> – also <a href="Ganze_Funktion" title="Ganze Funktion">ganz</a> – ist. Außerdem stimmt diese Definition mit der obigen Definition überein, falls die Distributionen kompakten Träger haben. Welche ganzen Funktionen hier als Fourier-Laplace-Transformationen auftreten können, charakterisiert der <a href="Satz_von_Paley-Wiener" title="Satz von Paley-Wiener">Satz von Paley-Wiener</a>.

</p>

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

<p>Für temperierte Distributionen kann man ebenfalls eine <a href="Laplace-Transformation" title="Laplace-Transformation">Laplace-Transformation</a> definieren. Diese sieht ähnlich aus wie die Fourier-Laplace-Transformation aus dem vorigen Abschnitt.

Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {S}}'}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle u\in {\mathcal {S}}'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/711571cd2af12304c8d6fa9ea236dfe13ad2ccc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.373ex; height:2.509ex;" alt="{\displaystyle u\in {\mathcal {S}}'}" loading="lazy"></span> eine temperierte Distribution mit Träger in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty [}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9414e096892e5858a261194f8b98d41e1a6793db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle [0,\infty [}" loading="lazy"></span>, dann ist die Laplace-Transformation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.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 u}" loading="lazy"></span> durch

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(u)(\xi ):=u(e^{-\langle \cdot ,\xi \rangle })}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42cdb5a9cfcda128d18b80d12f07a18b49ac7f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.983ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}(u)(\xi ):=u(e^{-\langle \cdot ,\xi \rangle })}" loading="lazy"></span></dd></dl>

<p>definiert. Das Resultat der Transformation ist ebenfalls wieder eine <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorphe Funktion</a>, die für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {Re}}(\xi )>0}">

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fcc0370836b22596b1d1b98957f0915ab0ca0ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.957ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {Re}}(\xi )>0}" loading="lazy"></span> definiert ist (sich aber eventuell auf eine größere Menge analytisch fortsetzen lässt). Im Gegensatz zur Fourier-Laplace-Transformation ist die Laplace-Transformation auch für temperierte Distributionen definiert, die keinen kompakten Träger haben. Dies ist möglich, da das Abklingverhalten von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\langle x,\xi \rangle }}">

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<annotation encoding="application/x-tex">{\displaystyle e^{-\langle x,\xi \rangle }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daf738e04128f20ff58b359043f090353d0673f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6ex; height:2.843ex;" alt="{\displaystyle e^{-\langle x,\xi \rangle }}" loading="lazy"></span> besser ist als das des Fourier-Kerns <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\mathrm {i} \langle x,\xi \rangle }}">

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<annotation encoding="application/x-tex">{\displaystyle e^{-\mathrm {i} \langle x,\xi \rangle }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d81ee43e51a08e4646968e5a27d25d381aaaf551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.457ex; height:2.843ex;" alt="{\displaystyle e^{-\mathrm {i} \langle x,\xi \rangle }}" loading="lazy"></span>.

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

<ul><li><a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Lars Hörmander</a>: <i>The Analysis of Linear Partial Differential Operators.</i> Band 1: <i>Distribution Theory and Fourier Analysis.</i> Second Edition. Springer-Verlag, Berlin u. a. 1990, ISBN 3-540-52345-6 (<i>Grundlehren der mathematischen Wissenschaften</i> 256).</li>

<li><a href="Otto_Forster" title="Otto Forster">Otto Forster</a>, Joachim Wehler: <i><a rel="nofollow" class="external text" href="http://www.pst.informatik.uni-muenchen.de/personen/wehler/wavelets10.PDF">Fourier-Transformation und Wavelets</a> (PDF; 575 kB).</i> 2001 (Skript).</li>

<li>R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie: <i>Fourier and Laplace transforms.</i> Cambridge University Press, 2003, ISBN 978-0-521-53441-3.</li>

<li><a href="Israel_Gelfand" class="mw-redirect" title="Israel Gelfand">Israel Gelfand</a>: <i>Verallgemeinerte Funktionen (Distributionen).</i> VEB Deutscher Verlag der Wissenschaften, Berlin (Ost).

<ul><li>Band 1: I. M. Gelfand, <a href="Georgi_Jewgenjewitsch_Schilow" title="Georgi Jewgenjewitsch Schilow">G. E. Schilow</a>: <i>Verallgemeinerte Funktionen und das Rechnen mit ihnen.</i> 1960 (<i><a href="Hochschulb%C3%BCcher_f%C3%BCr_Mathematik" title="Hochschulbücher für Mathematik">Hochschulbücher für Mathematik</a></i> 47, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220073-2842%22&key=cql">0073-2842</a></span></span>);</li>

<li>Band 2: I. M. Gelfand, G. E. Schilow: <i>Lineare topologische Räume, Räume von Grundfunktionen und verallgemeinerten Funktionen.</i> 1962 (<i>Hochschulbücher für Mathematik</i> 48);</li>

<li>Band 3: I. M. Gelfand, G. E. Schilow: <i>Einige Fragen zur Theorie der Differentialgleichungen.</i> 1964 (<i>Hochschulbücher für Mathematik</i> 49);</li>

<li>Band 4: I. M. Gelfand, <a href="Naum_Jakowlewitsch_Wilenkin" title="Naum Jakowlewitsch Wilenkin">N. J. Wilenkin</a>: <i>Einige Anwendungen der harmonischen Analyse. Gelfandsche Raumtripel.</i> 1964 (<i>Hochschulbücher für Mathematik</i> 50).</li>

<li>Band 5: I. M. Gelfand, M. I. Graev: <i>Integral geometry and representation theory</i> 1966, Academic Press.</li></ul></li>

<li>Klaus-Heinrich Peters: <i>Der Zusammenhang von Mathematik und Physik am Beispiel der Geschichte der Distributionen. Eine historische Untersuchung über die Grundlagen der Physik im Grenzbereich zu Mathematik, Philosophie und Kunst.</i> 2004 (Hamburg, Univ., Diss., 2003), <a rel="nofollow" class="external text" href="http://d-nb.info/972150358/34">online (PDF; 2,72 MB)</a>.</li></ul></div><div><div class="zim-footer">

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