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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Haar-Raum</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>Ein <b>Haar-Raum</b>, oder <b>Haarscher Raum</b> (benannt nach <a href="Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a>) wird in der <a href="Approximation" title="Approximation">Approximationstheorie</a> folgendermaßen definiert:
</p><p>Besitzen <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
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</math></span><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"></span> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängige</a>, auf einem <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</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,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span> <a href="Stetige_Funktion" title="Stetige Funktion">stetige Funktionen</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 g_{1},\dots ,g_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{1},\dots ,g_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ab736e4d81fa410e1a1f5971ce627bf1aa00a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.009ex;" alt="{\displaystyle g_{1},\dots ,g_{n}}" loading="lazy"></span> die Eigenschaft, dass <i>jedes</i> Element <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 \mathrm {span} \left\{g_{1},\dots ,g_{n}\right\},f\neq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
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<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">s</mi>
<mi mathvariant="normal">p</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">n</mi>
</mrow>
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
<mo>,</mo>
<mi>f</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {f}\in \mathrm {span} \left\{g_{1},\dots ,g_{n}\right\},f\neq 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49b97ba55be26eef5cb4ea9be7712cf543f05103.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.125ex; height:2.843ex;" alt="{\displaystyle {f}\in \mathrm {span} \left\{g_{1},\dots ,g_{n}\right\},f\neq 0}" 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 [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span> <i>höchstens</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 (n-1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df88c6333caaf6471cf277f24b802ff9931b133e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}" loading="lazy"></span> <a href="Nullstelle" title="Nullstelle">Nullstellen</a> hat, dann heißt die <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</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 U:=\mathrm {span} \left\{g_{1},\dots ,g_{n}\right\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">s</mi>
<mi mathvariant="normal">p</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">n</mi>
</mrow>
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
<mo>}</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U:=\mathrm {span} \left\{g_{1},\dots ,g_{n}\right\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a122a05dad0ce866cfd1a2b58123b77d5eee6e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.573ex; height:2.843ex;" alt="{\displaystyle U:=\mathrm {span} \left\{g_{1},\dots ,g_{n}\right\}}" loading="lazy"></span> Haar-Raum.
</p><p>Ein System solcher Funktionen <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_{1},\dots ,g_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{1},\dots ,g_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ab736e4d81fa410e1a1f5971ce627bf1aa00a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.009ex;" alt="{\displaystyle g_{1},\dots ,g_{n}}" loading="lazy"></span>, die einen Haar-Raum aufspannen, wird auch <i>Haarsches System</i> oder <i><a href="Pafnuti_Lwowitsch_Tschebyschow" title="Pafnuti Lwowitsch Tschebyschow">Tschebyschow</a>-System</i> genannt. Wird eine stetige Funktion durch Elemente eines Haar-Raumes approximiert, so existiert bezüglich der <a href="Maximumsnorm" title="Maximumsnorm">Maximumsnorm</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 \|\cdot \|_{\infty }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{\infty }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b723232adf7317abb1fc1c1326e1e4f79616a7e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.879ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{\infty }}" loading="lazy"></span> stets genau eine beste Approximation.
</p>
<div class="mw-heading mw-heading2"><h2 id="Interpolation_in_Haar-Räumen"><span id="Interpolation_in_Haar-R.C3.A4umen"></span>Interpolation in Haar-Räumen</h2></div>
<p>Hat man <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_{1},\dots ,x_{n}\in [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}\in [a,b]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47879bae04cbba0cfb552632efc8a65098f343c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.506ex; height:2.843ex;" alt="{\displaystyle x_{1},\dots ,x_{n}\in [a,b]}" loading="lazy"></span> paarweise verschiedene Punkte (Stützstellen) und Daten <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 y_{i},i=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{i},i=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00948fef8092f100bcd902c03d24f277d1dbb6c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.609ex; height:2.509ex;" alt="{\displaystyle y_{i},i=1,\dots ,n}" loading="lazy"></span>, so existiert genau ein <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 U}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>∈<!-- ∈ --></mo>
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\in U}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/081864b4dad059af21e4a80049e6025ffe390041.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.739ex; height:2.509ex;" alt="{\displaystyle g\in U}" loading="lazy"></span> 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 g(x_{i})=y_{i},i=1,\dots ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x_{i})=y_{i},i=1,\dots ,n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a3cb2a41eab93403c2fbbcecd29206574c2319d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.762ex; height:2.843ex;" alt="{\displaystyle g(x_{i})=y_{i},i=1,\dots ,n}" loading="lazy"></span>. Dies ist äquivalent zur Regularität der <a href="Vandermonde-Matrix" title="Vandermonde-Matrix">Vandermonde-Matrix</a>.
</p>
<dl><dd><b>Beweis</b> Bezeichne <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 \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
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<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1dfbdbc9f282a890a4539ce701c2029c1e1820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.364ex; height:2.843ex;" alt="{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}" loading="lazy"></span> den Körper, in den die Funktionen <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_{1},\dots ,g_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{1},\dots ,g_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ab736e4d81fa410e1a1f5971ce627bf1aa00a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.669ex; height:2.009ex;" alt="{\displaystyle g_{1},\dots ,g_{n}}" loading="lazy"></span> abbilden. Die 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 L\colon U\to \mathbb {K} ^{n},f\mapsto \left(f(x_{1}),...f(x_{n})\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>:<!-- : --></mo>
<mi>U</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msup>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle L\colon U\to \mathbb {K} ^{n},f\mapsto \left(f(x_{1}),...f(x_{n})\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50c311c73e0518840bde6aa64aa9b2a1ffd8b263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.02ex; height:2.843ex;" alt="{\displaystyle L\colon U\to \mathbb {K} ^{n},f\mapsto \left(f(x_{1}),...f(x_{n})\right)}" loading="lazy"></span> ist <a href="Lineare_Abbildung" title="Lineare Abbildung">linear</a>. Weil jedes <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 U}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∈<!-- ∈ --></mo>
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\in U}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/050aef6e2426a7c39a72282be5c11bb0fe7c0376.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.902ex; height:2.509ex;" alt="{\displaystyle f\in U}" loading="lazy"></span> höchstens <i>n-1</i> Nullstellen hat, ist der <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> der Abbildung nur die <a href="Nullfunktion" title="Nullfunktion">Nullfunktion</a>, d. h. L ist <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a>. Wegen <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 \dim \mathbb {K} ^{n}=n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>dim</mi>
<mo><!-- --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">K</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>=</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \dim \mathbb {K} ^{n}=n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68a7291d57b681b197298a3df8afb27ff6301258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.782ex; height:2.343ex;" alt="{\displaystyle \dim \mathbb {K} ^{n}=n}" loading="lazy"></span> ist L <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">surjektiv</a>, also insgesamt <a href="Bijektivit%C3%A4t" class="mw-redirect" title="Bijektivität">bijektiv</a>. Daraus folgt Existenz und Eindeutigkeit der Interpolationsfunktion g.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<ul><li>Der <a href="Vektorraum" title="Vektorraum">Vektorraum</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 \Pi _{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Π<!-- Π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Pi _{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09f95d02eefa1f08f4a994ebf0d60aad2c122dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle \Pi _{n}}" loading="lazy"></span> der <a href="Polynom" title="Polynom">Polynome</a> höchstens n-ten Grades ist ein Haar-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 \left\{1,x,\dots ,x^{n}\right\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mi>x</mi>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>}</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{1,x,\dots ,x^{n}\right\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e55aea0090ae0c730e98fe3800798edbe51de7a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.577ex; height:2.843ex;" alt="{\displaystyle \left\{1,x,\dots ,x^{n}\right\}}" loading="lazy"></span> ist ein Haarsches System.</li>
<li>Das System <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\{x,\dots ,x^{n}\right\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mo>}</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{x,\dots ,x^{n}\right\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0da60412fedafc61cbff2680c209318b1aa71d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.381ex; height:2.843ex;" alt="{\displaystyle \left\{x,\dots ,x^{n}\right\}}" loading="lazy"></span> ist jedoch kein Haarscher Raum.</li>
<li>Die <a href="Trigonometrisches_Polynom" title="Trigonometrisches Polynom">trigonometrischen Polynome</a> bilden ein Haar-Raum mit Haarschem System <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\{1,e^{ix},\dots ,e^{i(n-1)x}\right\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>{</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>x</mi>
</mrow>
</msup>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>}</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{1,e^{ix},\dots ,e^{i(n-1)x}\right\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93602a2a62a0a16fd4c55c3dfd50406302c246eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.488ex; height:4.843ex;" alt="{\displaystyle \left\{1,e^{ix},\dots ,e^{i(n-1)x}\right\}}" loading="lazy"></span> (Polynome 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 e^{ix}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>x</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{ix}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5271aa5bef13f6bd715afeda45bc59ae37d7c6d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.823ex; height:2.676ex;" alt="{\displaystyle e^{ix}}" loading="lazy"></span>).</li>
<li><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\{1,\sin(x),\cos(x),\dots ,\sin(nx),\cos(nx)\right\},\left\{1,\sin(x),\dots ,\sin(nx)\right\},\left\{1,\cos(x),\dots ,\cos(nx)\right\},\;x\in [0,2\pi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mo>{</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>}</mo>
</mrow>
<mo>,</mo>
<mrow>
<mo>{</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>}</mo>
</mrow>
<mo>,</mo>
<mrow>
<mo>{</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>}</mo>
</mrow>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\{1,\sin(x),\cos(x),\dots ,\sin(nx),\cos(nx)\right\},\left\{1,\sin(x),\dots ,\sin(nx)\right\},\left\{1,\cos(x),\dots ,\cos(nx)\right\},\;x\in [0,2\pi )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc3ea66429eb412479d97beea755ffe99b0b4029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.045ex; height:2.843ex;" alt="{\displaystyle \left\{1,\sin(x),\cos(x),\dots ,\sin(nx),\cos(nx)\right\},\left\{1,\sin(x),\dots ,\sin(nx)\right\},\left\{1,\cos(x),\dots ,\cos(nx)\right\},\;x\in [0,2\pi )}" loading="lazy"></span> sind jeweils Haarsche Systeme.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Historie">Historie</h2></div>
<p>Erstmals formulierte und bewies Haar die <i>Haar condition</i> 1918 in: <i>Die Minkowskische Geometrie und die Annäherung an stetige Funktionen</i>, <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>, Band 78, S. 294–311. Andere Beweise formulierten Vlastimil Pták 1958 (<i>A remark on approximation of continuous functions</i> in Czechoslovak Math. Journal, Band 8, S. 251–256) und <a href="Isadore_M._Singer" title="Isadore M. Singer">Singer</a> 1960 (<i>On best approximation of continuous functions</i> in Mathematische Annalen, Band 140, S. 165–168).<sup id="cite_ref-Cheney_1-0" class="reference"><a href="#cite_note-Cheney-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>Günther Hämmerlin, Karl-Heinz Hoffmann: <i>Numerische Mathematik.</i> Springer, Berlin 1994, ISBN 3-540-58033-6.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-Cheney-1"><span class="mw-cite-backlink"><a href="#cite_ref-Cheney_1-0">↑</a></span> <span class="reference-text">
Elliot Ward Cheney: <i>Introduction to Approximation Theory.</i> McGraw-Hill Book Company, 1966, <a href="Library_of_Congress_Catalog_Card_Number" class="mw-redirect" title="Library of Congress Catalog Card Number">Library of Congress Catalog Card Number</a> 65-25916, ISBN 0-07-010757-2, S. 227 + 242 + 248 + 251</span>
</li>
</ol></div><!--htdig_noindex--><div><div class="zim-footer">
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