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<title>Proximum</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Proximum</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">Das Proximum (oder auch Bestapproximation) ist ein vor allem in der <a href="Numerik" class="mw-redirect" title="Numerik">numerischen Mathematik</a> verwendeter Begriff aus der Theorie der <a href="Metrischer_Raum" title="Metrischer Raum">metrischen Räume</a>. Das Proximum zu einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> nicht enthaltenden Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> ist derjenige Punkt aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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

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

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d)}">

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<annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}" loading="lazy"> ein <a href="Metrischer_Raum" title="Metrischer Raum">metrischer Raum</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22ed7bcf8cec0dba504a57e067b74f9f85f1ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y\subset X}" loading="lazy"> eine <a href="Teilmenge" title="Teilmenge">Teilmenge</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> beliebig. Der Abstand des Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> wird mittels der <a href="Distanzfunktion" title="Distanzfunktion">Distanzfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dist} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/911f6e57962b06983281ec0e63ebbb426f8a9ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.76ex; height:2.176ex;" alt="{\displaystyle \operatorname {dist} }" loading="lazy"> definiert durch

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {dist} (x,Y):=\inf _{y\in Y}d(x,y)\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0e637232b6f80cb2299ed7517418b9e9b077a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.584ex; height:4.343ex;" alt="{\displaystyle \operatorname {dist} (x,Y):=\inf _{y\in Y}d(x,y)\,.}" loading="lazy"></dd></dl>

Existiert nun ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8d2bb59bff6632a582019d11ddca16d46544f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.873ex; height:2.509ex;" alt="{\displaystyle p\in Y}" loading="lazy"> mit:

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<annotation encoding="application/x-tex">{\displaystyle d(x,p)=\operatorname {dist} (x,Y)\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56ed0a90d1480dc7aedb3f96f330a2e048e9e4e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.75ex; height:2.843ex;" alt="{\displaystyle d(x,p)=\operatorname {dist} (x,Y)\,}" loading="lazy"></dd></dl>

so nennt man <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> Proximum oder Bestapproximation zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">.

Wenn ein Proximum existiert, so muss es nicht eindeutig sein.

Üblicherweise hat man es in der <a href="Approximation" title="Approximation">Approximationstheorie</a> mit einem <a href="Normierter_Raum" title="Normierter Raum">normierten Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lVert \cdot \rVert )}">

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<annotation encoding="application/x-tex">{\displaystyle (X,\lVert \cdot \rVert )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4b135498472d07063403a1a2fe311df8c2e8029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.795ex; height:2.843ex;" alt="{\displaystyle (X,\lVert \cdot \rVert )}" loading="lazy"> zu tun. Ein Proximum <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22ed7bcf8cec0dba504a57e067b74f9f85f1ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y\subset X}" loading="lazy"> ist dann – falls existent – charakterisiert durch die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle \lVert x-p\rVert =\inf _{y\in Y}\lVert x-y\rVert }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db823d68a400c8995a2f8a759ddcf2b2a1fb0a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.581ex; height:4.343ex;" alt="{\displaystyle \lVert x-p\rVert =\inf _{y\in Y}\lVert x-y\rVert }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Zur_Existenz_eines_Proximums">Zur Existenz eines Proximums</h2></div>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (X,d)\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cea67d2aa5fe3c3968ce00c9b6294c01cf8cd13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.426ex; height:2.843ex;" alt="{\displaystyle (X,d)\,}" loading="lazy"> ein <a href="Metrischer_Raum" title="Metrischer Raum">metrischer Raum</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/826569be03f873b81cdc6f12637ef5520c369d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}" loading="lazy"> sei eine <a href="Kompakter_Raum" title="Kompakter Raum">kompakte</a> Teilmenge. Dann hat jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> ein Proximum in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><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">.</li></ul>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,\lVert \cdot \rVert )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4b135498472d07063403a1a2fe311df8c2e8029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.795ex; height:2.843ex;" alt="{\displaystyle (X,\lVert \cdot \rVert )}" loading="lazy"> ein normierter Raum. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subset X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5e2e950065e42f0b8bd69fdf32212e3dfb14855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.866ex; height:2.176ex;" alt="{\displaystyle V\subset X}" loading="lazy"> sei ein endlichdimensionaler Teilraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset V}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5adb2179f29fa8c205b877c948bc14b74a18a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.659ex; height:2.176ex;" alt="{\displaystyle Y\subset V}" loading="lazy"> eine abgeschlossene Teilmenge. Dann hat jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> ein Proximum in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Eindeutigkeit_des_Proximums_in_Tschebyschow-Systemen">Eindeutigkeit des Proximums in Tschebyschow-Systemen</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C[a,b],U\subset C[a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C[a,b],U\subset C[a,b]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77a752d698b4aff960db039fd609ace1e19314b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.677ex; height:2.843ex;" alt="{\displaystyle f\in C[a,b],U\subset C[a,b]}" loading="lazy"> ein <a href="Tschebyschow-System" class="mw-redirect" title="Tschebyschow-System">Tschebyschow-System</a>. Dann ist das Proximum für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> eindeutig bestimmt.

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

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> ein endlichdimensionaler Unterraum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle C[a,b]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c1625217aad8c105c50c975599e45192b2bfbec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.321ex; height:2.843ex;" alt="{\displaystyle C[a,b]}" loading="lazy">. Ist für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C[a,b]}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C[a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18a75183a73efed5373e1d10284bceb0c4e48741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle f\in C[a,b]}" loading="lazy"> das Proximum aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> eindeutig bestimmt, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> ein <a href="Tschebyschow-System" class="mw-redirect" title="Tschebyschow-System">Tschebyschow-System</a>.

<div class="mw-heading mw-heading2"><h2 id="Alternanten-Kriterium_in_Tschebyschow-Systemen">Alternanten-Kriterium in Tschebyschow-Systemen</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C[a,b],U\subset C[a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C[a,b],U\subset C[a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77a752d698b4aff960db039fd609ace1e19314b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.677ex; height:2.843ex;" alt="{\displaystyle f\in C[a,b],U\subset C[a,b]}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionales <a href="Tschebyschow-System" class="mw-redirect" title="Tschebyschow-System">Tschebyschow-System</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0}\in U}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/133e383f80a11def53dfb9316d0d4f71d32b20da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.509ex;" alt="{\displaystyle u_{0}\in U}" loading="lazy"> ist genau dann ein Proximum für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, wenn es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n}\leq b}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n}\leq b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5143cfd3bd4c3442a6c6e7538a1d4a8cbd11eea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.759ex; height:2.509ex;" alt="{\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n}\leq b}" loading="lazy"> gibt, so dass

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x_{i})-u_{0}(x_{i})|=\max _{x\in [a,\,b]}|f(x)-u_{0}(x)|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6adf866800e476d74129c9e03572549553593b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.605ex; height:4.509ex;" alt="{\displaystyle |f(x_{i})-u_{0}(x_{i})|=\max _{x\in [a,\,b]}|f(x)-u_{0}(x)|}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0,\ldots ,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=0,\ldots ,n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/063c205352931a55f96dcc9aef7ada666356d9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=0,\ldots ,n}" loading="lazy"> (<a href="Extremalpunkt" title="Extremalpunkt">Extremalpunkt</a>)</li>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sign} \left(f(x_{i-1})-u_{0}(x_{i-1})\right)=-\operatorname {sign} (f(x_{i})-u_{0}(x_{i}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee583d38f2bdd855ec938bfb9d77a3ed81445523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.91ex; height:2.843ex;" alt="{\displaystyle \operatorname {sign} \left(f(x_{i-1})-u_{0}(x_{i-1})\right)=-\operatorname {sign} (f(x_{i})-u_{0}(x_{i}))}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5726d00b79af1b4666a6319c45381579dc85a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,n}" loading="lazy"> (alternierend)</li></ul>

Dies folgt aus dem <a href="Kolmogorow" class="mw-redirect" title="Kolmogorow">Kolmogorow</a>-Kriterium aus der <a href="Approximationstheorie" class="mw-redirect" title="Approximationstheorie">Approximationstheorie</a>. Auf diesem Kriterium basiert der <a href="Remez-Algorithmus" title="Remez-Algorithmus">Remez-Algorithmus</a> zur numerischen Bestimmung des Proximums in Tschebyschow-Systemen.

<div class="mw-heading mw-heading2"><h2 id="Proximum_im_Hilbertraum">Proximum im Hilbertraum</h2></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ein <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22ed7bcf8cec0dba504a57e067b74f9f85f1ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y\subset X}" loading="lazy"> eine abgeschlossene konvexe nichtleere Teilmenge, dann ist das Proximum eindeutig, das heißt, es existiert zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> genau ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8d2bb59bff6632a582019d11ddca16d46544f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.873ex; height:2.509ex;" alt="{\displaystyle p\in Y}" loading="lazy"> mit

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lVert x-p\rVert \leq \lVert x-y\rVert \,\,\forall \,y\in Y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d93413802e2c34228c874ec18d19fd063634cabe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.637ex; height:2.843ex;" alt="{\displaystyle \lVert x-p\rVert \leq \lVert x-y\rVert \,\,\forall \,y\in Y}" loading="lazy">.</dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> ein abgeschlossener <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a>, so erhält man das Proximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> als <a href="Orthogonalprojektion" title="Orthogonalprojektion">Orthogonalprojektion</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">.

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

<ul><li><a href="Alternantensatz" title="Alternantensatz">Alternantensatz</a></li>

<li><a href="Minimall%C3%B6sung" title="Minimallösung">Minimallösung</a></li></ul>

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

<ul><li>Arnold Schönhage: Approximationstheorie. de Gruyter, Berlin 1971, ISBN 3-11-001982-5 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=XENFGizjTkUC">eingeschränkte Vorschau</a> in der Google-Buchsuche).</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-05-24" href="https://de.wikipedia.org/wiki/?title=Proximum&oldid=256296219">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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