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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hermitesche Funktion</span></h1>
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<p>Die <b>Hermiteschen Funktionen</b> <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 h_{n}(x)}">
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<annotation encoding="application/x-tex">{\displaystyle h_{n}(x)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2746146c2ba2fbbe31925bb3a1eb6af968c7de94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.696ex; height:2.843ex;" alt="{\displaystyle h_{n}(x)}" loading="lazy"></span> erhält man aus den <a href="Hermitesches_Polynom" title="Hermitesches Polynom">Hermiteschen Polynomen</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 H_{n}(x)}">
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<annotation encoding="application/x-tex">{\displaystyle H_{n}(x)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505cd70a83ef6433715abc22c4d2ed86058c2738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.289ex; height:2.843ex;" alt="{\displaystyle H_{n}(x)}" loading="lazy"></span>, indem man diese mit der Dichte der <a href="Normalverteilung" title="Normalverteilung">Gaußschen Normalverteilung</a> multipliziert.
</p>
<dl><dd><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 h_{n}(x)={\frac {(-1)^{n}}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}e^{x^{2}/2}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}}={\frac {1}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}H_{n}(x)e^{-{\frac {1}{2}}x^{2}},}">
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<annotation encoding="application/x-tex">{\displaystyle h_{n}(x)={\frac {(-1)^{n}}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}e^{x^{2}/2}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}}={\frac {1}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}H_{n}(x)e^{-{\frac {1}{2}}x^{2}},}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddae814dbe66753c8dfe9c2f6aecb42bf2a9b50e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:60.173ex; height:8.509ex;" alt="{\displaystyle h_{n}(x)={\frac {(-1)^{n}}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}e^{x^{2}/2}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}e^{-x^{2}}={\frac {1}{\sqrt {2^{n}n!{\sqrt {\pi }}}}}H_{n}(x)e^{-{\frac {1}{2}}x^{2}},}" loading="lazy"></span></dd>
<dd><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 \int _{-\infty }^{\infty }h_{n}(x)h_{m}(x)\,\mathrm {d} x=\delta _{n,m}\qquad \qquad n,m=0,1,2,\ldots }">
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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }h_{n}(x)h_{m}(x)\,\mathrm {d} x=\delta _{n,m}\qquad \qquad n,m=0,1,2,\ldots }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7b7894b97eb5c68c499038fb38e29bf1c1a82b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.112ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }h_{n}(x)h_{m}(x)\,\mathrm {d} x=\delta _{n,m}\qquad \qquad n,m=0,1,2,\ldots }" loading="lazy"></span></dd></dl>
<p>Sie sind ein sehr gutes Beispiel für die Definition (Erzeugung) einer <a href="Orthonormalit%C3%A4t" class="mw-redirect" title="Orthonormalität">orthonormalen</a> <a href="Hilbertbasis" class="mw-redirect" title="Hilbertbasis">Basis</a>, ähnlich der <a href="Sinus" class="mw-redirect" title="Sinus">Sinus</a>-/<a href="Kosinus" class="mw-redirect" title="Kosinus">Kosinusfunktionen</a>. Während letztere in der Lage sind, mittels der Spektralanalyse (<a href="Fourieranalyse" class="mw-redirect" title="Fourieranalyse">Fourieranalyse</a>) ein periodisches Signal in ein Frequenzspektrum zu zerlegen, erlauben die Hermiteschen Funktionen die Beschreibung singulärer Ereignisse.
</p><p>Eine wichtige Bedeutung haben sie in der Physik zur Konstruktion der <a href="Orthonormalit%C3%A4t" class="mw-redirect" title="Orthonormalität">orthonormierten</a> Lösungsfunktionen des
<a href="Harmonischer_Oszillator_(Quantenmechanik)" title="Harmonischer Oszillator (Quantenmechanik)">quantenmechanischen harmonischen Oszillators</a>. Motiviert durch die <a href="Harmonischer_Oszillator_(Quantenmechanik)#Die_Leiteroperatormethode" title="Harmonischer Oszillator (Quantenmechanik)">Erzeugungs- und Vernichtungsoperatoren</a> der Quantenmechanik erhält man folgende rekursive Darstellung der hermiteschen Funktionen
</p>
<dl><dd><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 h_{n}(x)=n^{-{\frac {1}{2}}}a^{\dagger }h_{n-1}(x),\qquad h_{0}(x)=\pi ^{-{\frac {1}{4}}}e^{-{\frac {1}{2}}x^{2}},}">
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<annotation encoding="application/x-tex">{\displaystyle h_{n}(x)=n^{-{\frac {1}{2}}}a^{\dagger }h_{n-1}(x),\qquad h_{0}(x)=\pi ^{-{\frac {1}{4}}}e^{-{\frac {1}{2}}x^{2}},}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e8f1f751fa98dc39ce9a01edc9b45d6dbabc76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.368ex; height:4.009ex;" alt="{\displaystyle h_{n}(x)=n^{-{\frac {1}{2}}}a^{\dagger }h_{n-1}(x),\qquad h_{0}(x)=\pi ^{-{\frac {1}{4}}}e^{-{\frac {1}{2}}x^{2}},}" loading="lazy"></span></dd></dl>
<p>dabei ist der Operator <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^{\dagger }}">
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<dl><dd><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^{\dagger }={\frac {1}{\sqrt {2}}}{\Big (}x-{\frac {\mathrm {d} }{\mathrm {d} x}}{\Big )}.}">
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<annotation encoding="application/x-tex">{\displaystyle a^{\dagger }={\frac {1}{\sqrt {2}}}{\Big (}x-{\frac {\mathrm {d} }{\mathrm {d} x}}{\Big )}.}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52753c8ce188ebee5505010dc8de0c040b303003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.276ex; height:6.343ex;" alt="{\displaystyle a^{\dagger }={\frac {1}{\sqrt {2}}}{\Big (}x-{\frac {\mathrm {d} }{\mathrm {d} x}}{\Big )}.}" loading="lazy"></span></dd></dl>
<p>Singuläre Ereignisse werden in der Regel durch <a href="Intensit%C3%A4t_(Physik)" title="Intensität (Physik)">Intensität</a>, <a href="Mittelwert" title="Mittelwert">Mittelwert</a> und <a href="Empirische_Standardabweichung" class="mw-redirect" title="Empirische Standardabweichung">Standardabweichung</a> charakterisiert. Diese Kennwerte können aber für verschiedene, sehr unterschiedliche Ereignisse identisch sein, so dass sie für die Charakterisierung nicht ausreichen. Daher bestimmt man die sogenannten „höheren statistischen <a href="Moment_(Stochastik)" title="Moment (Stochastik)">Momente</a>“ als weitere Vergleichsgrößen. Diese sind jedoch sehr empfindlich auf <a href="Rauschen_(Physik)" title="Rauschen (Physik)">Rauschen</a> und Drift der Nulllinie und daher nur bedingt geeignet. Entwickelt man eine Verteilung in Hermiteschen Funktionen, so sind die Koeffizienten sehr stabil, da die Funktionen nur im zentralen Bereich leben und somit weiter außenliegende Messdaten geeignet dämpfen.
</p><p>Die Entwicklung einer ein Ereignis repräsentierenden Funktion nach Hermiteschen Funktionen hat eine gewisse Ähnlichkeit mit der <a href="Wavelet-Transformation" title="Wavelet-Transformation">Wavelet-Transformation</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Hermitesche_Funktionen_als_Eigenfunktionen_der_Fourier-Transformation">Hermitesche Funktionen als Eigenfunktionen der Fourier-Transformation</h2></div>
<p>Die Hermiteschen Funktionen sind <a href="Eigenfunktion" class="mw-redirect" title="Eigenfunktion">Eigenfunktionen</a> der <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a> im Eindimensionalen 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 \left(-\mathrm {i} \right)^{n}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle \left(-\mathrm {i} \right)^{n}}</annotation>
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</math></span><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"></span>:
</p>
<dl><dd><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}}\,h_{n}=\left(-\mathrm {i} \right)^{n}\,h_{n}\qquad \left(n\in \mathbb {N} _{0}\right).}">
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\,h_{n}=\left(-\mathrm {i} \right)^{n}\,h_{n}\qquad \left(n\in \mathbb {N} _{0}\right).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190bd961ac51d3511deda736406e2d807ab8120d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.24ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\,h_{n}=\left(-\mathrm {i} \right)^{n}\,h_{n}\qquad \left(n\in \mathbb {N} _{0}\right).}" loading="lazy"></span></dd></dl>
<p>Mehr noch, sie bilden ferner im <a href="Lp-Raum#Der_Hilbertraum_L2" title="Lp-Raum">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 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></span><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"></span></a> ein <a href="Vollst%C3%A4ndiges_Orthonormalsystem" class="mw-redirect" title="Vollständiges Orthonormalsystem">vollständiges Orthonormalsystem</a> von Eigenfunktionen.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>I. N. Bronstein, K. A. Semendjajew (Begründer), Günter Grosche (Bearb.), <a href="Eberhard_Zeidler_(Mathematiker)" title="Eberhard Zeidler (Mathematiker)">Eberhard Zeidler</a> (Hrsg.): <i><a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Teubner-Taschenbuch der Mathematik</a></i>. Teubner, Stuttgart 1996, ISBN 3-8154-2001-6.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Helmut Fischer, Helmut Kaul: <i>Mathematik für Physiker, Band 2: Gewöhnliche und partielle Differentialgleichungen, mathematische Grundlagen der Quantenmechanik</i>. 2.&nbsp;Aufl., B.G. Teubner, Wiesbaden 2004. ISBN 3-519-12080-1, §12 Abschn.&nbsp;4.2, S.&nbsp;300–301.</span>
</li>
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