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<title>Poisson-Transformation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Poisson-Transformation</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>In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist die <b>Poisson-Transformation</b> ein Verfahren zur Konstruktion <a href="Harmonische_Funktion" title="Harmonische Funktion">harmonischer Funktionen</a> auf der <a href="Einheitskreisscheibe" class="mw-redirect" title="Einheitskreisscheibe">Einheitskreisscheibe</a>. Das Integral, das in dieser Konstruktion auftaucht, heißt <b>Poisson-Integral</b><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> und der <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a> dessen wird <b>Poisson-Kern</b> genannt.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Benannt sind sowohl die Transformation, das Integral und der Integralkern nach dem Mathematiker und Physiker <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>.

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

<p>Gegeben ist eine (beschränkte) Funktion auf dem <a href="Einheitskreis" title="Einheitskreis">Einheitskreis</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 S^{1}=\partial D^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{1}=\partial D^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfa8ed5fc215d5ec61790fce829f36f2cd684fae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.971ex; height:2.676ex;" alt="{\displaystyle S^{1}=\partial D^{2}}" loading="lazy"></span>, gesucht wird eine (beschränkte) <i>harmonische</i> Funktion auf der Einheitskreisscheibe <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 D^{2}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy"></span>, deren Werte auf dem Rand mit der gegebenen Funktion <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 f}">

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

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</math></span><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"></span> übereinstimmen.

</p><p>Mit anderen Worten: es soll das <a href="Dirichlet-Problem" class="mw-redirect" title="Dirichlet-Problem">Dirichlet-Problem</a> für die <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b7a4ca4af9242e566083c9187f695379aedd8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.875ex; height:2.176ex;" alt="{\displaystyle \Delta \Phi =0}" loading="lazy"></span></dd></dl>

<p>auf der Kreisscheibe gelöst werden.

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

<p>Der <i>Poisson-Kern</i> ist die durch

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<annotation encoding="application/x-tex">{\displaystyle P(x,\xi )={\frac {1-\|x\|^{2}}{\|\xi -x\|^{2}}}\ \forall x\in D^{2},\xi \in S^{1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/860d94e997989e7901edde22c3cacf0df1e31422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.097ex; height:6.676ex;" alt="{\displaystyle P(x,\xi )={\frac {1-\|x\|^{2}}{\|\xi -x\|^{2}}}\ \forall x\in D^{2},\xi \in S^{1}}" loading="lazy"></span></dd></dl>

<p>gegebene Funktion.

</p><p>Die <i>Poisson-Transformation</i> ist die <a href="Integraltransformation" class="mw-redirect" title="Integraltransformation">Integraltransformation</a> mit <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</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 P}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>: einer Funktion <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 f\in L^{\infty }(S^{1})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e8b47d0df571cf7ce911c1a64c643c422aa7cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.963ex; height:3.176ex;" alt="{\displaystyle f\in L^{\infty }(S^{1})}" loading="lazy"></span> wird die 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 D^{2}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy"></span> definierte Funktion

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<annotation encoding="application/x-tex">{\displaystyle Pf(x)=\int _{S^{1}}f(\xi )P(x,\xi )d\sigma (\xi )\ \forall x\in D^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac3d2d5c239e8b09b6a9d414ff11679c9bb7a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.554ex; height:5.676ex;" alt="{\displaystyle Pf(x)=\int _{S^{1}}f(\xi )P(x,\xi )d\sigma (\xi )\ \forall x\in D^{2}}" loading="lazy"></span></dd></dl>

<p>zugeordnet, wobei <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 d\sigma }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a5824bc315231366cb8a7268d5143a62d59a178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle d\sigma }" loading="lazy"></span> das uniforme <a href="Wahrscheinlichkeitsma%C3%9F" title="Wahrscheinlichkeitsmaß">Wahrscheinlichkeitsmaß</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 S^{1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"></span> bezeichnet.

</p><p>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 Pf}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ba40ee0dac66ee1c2cdd22e05a7ef528f2ccb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle Pf}" loading="lazy"></span> eine <a href="Beschr%C3%A4nkte_Funktion" class="mw-redirect" title="Beschränkte Funktion">beschränkte</a> harmonische Funktion ist.

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

<p>Die Poisson-Transformation stellt eine <a href="Bijektion" class="mw-redirect" title="Bijektion">Bijektion</a> zwischen der Menge der beschränkten Funktionen 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 S^{1}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"></span> und der Menge der beschränkten harmonischen Funktionen 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 D^{2}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy"></span> her.

</p><p>Mit anderen Worten: zu jeder Funktion <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 f\in L^{\infty }(S^{1})}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e8b47d0df571cf7ce911c1a64c643c422aa7cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.963ex; height:3.176ex;" alt="{\displaystyle f\in L^{\infty }(S^{1})}" loading="lazy"></span> gibt es eine eindeutige <i>harmonische</i> Funktion <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 g\in L^{\infty }(D^{2})}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle g\in L^{\infty }(D^{2})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e09a30acd9338a3b35f5da044640ab2107bbce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.203ex; height:3.176ex;" alt="{\displaystyle g\in L^{\infty }(D^{2})}" loading="lazy"></span> mit <a href="Randwert" class="mw-redirect" title="Randwert">Randwerten</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 f}">

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

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</math></span><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"></span>.

</p><p>Die Bijektion erhält die <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^{\infty }}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9ab400cc4dfd865180cd84c72dc894ca457671f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}" loading="lazy"></span>-Norm.

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

<p>Die Poisson-Transformation lässt sich auf die n-dimensionale <a href="Einheitskugel" title="Einheitskugel">Einheitskugel</a> verallgemeinern, in diesem Fall ist der Poisson-Kern <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(x,\xi )={\tfrac {1-\|x\|^{2}}{\|\xi -x\|^{n}}}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1599770c2d6a9095f24f108a58eb9eb77ea797b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.439ex; height:5.176ex;" alt="{\displaystyle P(x,\xi )={\tfrac {1-\|x\|^{2}}{\|\xi -x\|^{n}}}}" loading="lazy"></span> 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 x\in D^{n},\xi \in \partial D^{n}=S^{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in D^{n},\xi \in \partial D^{n}=S^{n-1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b8abf46d9ae3692b47f1e821ba3b9a64e70d529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.618ex; height:3.009ex;" alt="{\displaystyle x\in D^{n},\xi \in \partial D^{n}=S^{n-1}}" loading="lazy"></span>.

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

<ul><li>Helgason, Sigurdur: <i>Topics in harmonic analysis on homogeneous spaces.</i> Progress in Mathematics, 13. Birkhäuser, Boston, Mass., 1981. ISBN 3-7643-3051-1</li>

<li>Quint, J.-F.: <i>An overview of Patterson-Sullivan theory</i>, Workshop "The barycenter method", FIM, Zurich, May 2006 <a rel="nofollow" class="external text" href="https://www.math.u-bordeaux.fr/~jquint/publications/courszurich.pdf">(Online)</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Poisson-Integral</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, ISBN 3-8274-0439-8.<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:Poisson-Transformation&rft.atitle=Poisson-Integral&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Poisson-Kern</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, ISBN 3-8274-0439-8.<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:Poisson-Transformation&rft.atitle=Poisson-Kern&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

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