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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Multirésolution</span></h1>
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<p>En mathématiques, une <b>approximation multirésolution</b> désigne une <a href="Suite_(math%C3%A9matiques)" title="Suite (mathématiques)">suite</a> de <a href="Sous-espace_vectoriel" title="Sous-espace vectoriel">sous-espaces vectoriels</a> vérifiant un ensemble de caractéristiques.
</p>
<div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2></div>
<p>Une <a href="Suite_(math%C3%A9matiques)" title="Suite (mathématiques)">suite</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 (V_{j})_{j\in \mathbb {Z} }}">
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<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
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<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle (V_{j})_{j\in \mathbb {Z} }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97d1cc793ea6e203c4450a3cc2368d1be2dadb2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.176ex; height:3.009ex;" alt="{\displaystyle (V_{j})_{j\in \mathbb {Z} }}" loading="lazy"></span> de <a href="Sous-espace_vectoriel" title="Sous-espace vectoriel">sous-espaces vectoriels</a> <a href="Ferm%C3%A9_(topologie)" title="Fermé (topologie)">fermés</a> de <a href="Espace_L2" title="Espace L2"><span class="texhtml">L<sup>2</sup>(<b>R</b>)</span></a> est une approximation multirésolution si elle vérifie les cinq propriétés suivantes<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>&nbsp;:
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<ul><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 \forall j\in \mathbb {Z} \quad V_{j+1}\subset V_{j}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>j</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
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<mspace width="1em"></mspace>
<msub>
<mi>V</mi>
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<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<annotation encoding="application/x-tex">{\displaystyle \forall j\in \mathbb {Z} \quad V_{j+1}\subset V_{j}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ac51f4fef02bd2a570cd81b95f42e4e2cd0b948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.693ex; height:2.843ex;" alt="{\displaystyle \forall j\in \mathbb {Z} \quad V_{j+1}\subset V_{j}}" 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 {\overline {\bigcup _{j\in \mathbb {Z} }V_{j}}}=L^{2}(\mathbb {R} ){\text{ et }}\bigcap _{j\in \mathbb {Z} }V_{j}=\{0\}}">
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mtext>&nbsp;et&nbsp;</mtext>
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle {\overline {\bigcup _{j\in \mathbb {Z} }V_{j}}}=L^{2}(\mathbb {R} ){\text{ et }}\bigcap _{j\in \mathbb {Z} }V_{j}=\{0\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/424bd5c702b3c5fe6f417adca29a48e7dde2623b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.453ex; height:6.509ex;" alt="{\displaystyle {\overline {\bigcup _{j\in \mathbb {Z} }V_{j}}}=L^{2}(\mathbb {R} ){\text{ et }}\bigcap _{j\in \mathbb {Z} }V_{j}=\{0\}}" 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 \forall j\in \mathbb {Z} \quad f\in V_{j}\iff f(2\cdot )\in V_{j+1}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
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<mi>V</mi>
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<annotation encoding="application/x-tex">{\displaystyle \forall j\in \mathbb {Z} \quad f\in V_{j}\iff f(2\cdot )\in V_{j+1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ae82225b984c2308bc5b13c64c3b4dce31d85f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.349ex; height:3.009ex;" alt="{\displaystyle \forall j\in \mathbb {Z} \quad f\in V_{j}\iff f(2\cdot )\in V_{j+1}}" 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 \forall (j,k)\in \mathbb {Z} ^{2}\quad f\in V_{j}\iff f(\cdot -2^{-j}k)\in V_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mspace width="1em"></mspace>
<mi>f</mi>
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<mi>V</mi>
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<mi>j</mi>
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<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="thickmathspace"></mspace>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>−<!-- − --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>j</mi>
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</msup>
<mi>k</mi>
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<mo>∈<!-- ∈ --></mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall (j,k)\in \mathbb {Z} ^{2}\quad f\in V_{j}\iff f(\cdot -2^{-j}k)\in V_{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c1c873880fd36da1894199c044f4e7c55f16d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.597ex; height:3.343ex;" alt="{\displaystyle \forall (j,k)\in \mathbb {Z} ^{2}\quad f\in V_{j}\iff f(\cdot -2^{-j}k)\in V_{j}}" loading="lazy"></span></li>
<li>Il existe <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 \theta \in V_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>∈<!-- ∈ --></mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \theta \in V_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa791cc9b288ccfccba69fb47965c08458ea6b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.341ex; height:2.509ex;" alt="{\displaystyle \theta \in V_{0}}" loading="lazy"></span> tel que <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(\theta (\cdot -n)_{n\in \mathbb {Z} }\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>(</mo>
<mrow>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo>−<!-- − --></mo>
<mi>n</mi>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
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<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left(\theta (\cdot -n)_{n\in \mathbb {Z} }\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/771d085b800bcaf48bc55efa2dd5f57ff70f834f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.002ex; height:2.843ex;" alt="{\displaystyle \left(\theta (\cdot -n)_{n\in \mathbb {Z} }\right)}" loading="lazy"></span> soit une <a href="Suite_de_Riesz" title="Suite de Riesz">base de Riesz</a> de <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 V_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle V_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae15ff9b845587dc4e1816f59c3fed0e71a132f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2></div>
<div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Mallat1989"><span class="ouvrage" id="Stéphane_Mallat1989"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="St%C3%A9phane_Mallat" title="Stéphane Mallat">Stéphane Mallat</a>, «&nbsp;<cite style="font-style:normal" lang="en">Multiresolution approximations and wavelet orthonormal bases of <span class="texhtml">L<sup>2</sup>(<b>R</b>)</span></cite>&nbsp;», <i><span class="lang-en" lang="en"><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Trans. Amer. Math. Soc.</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr>&nbsp;315, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr>&nbsp;1,‎ <time>1989</time>, <abbr class="abbr" title="pages">p.</abbr>&nbsp;<span class="nowrap">69-87</span> <small style="line-height:1em;">(<a href="Digital_Object_Identifier" title="Digital Object Identifier">DOI</a>&nbsp;<span class=" noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/2001373">10.2307/2001373</a></span>, <a rel="nofollow" class="external text" href="http://www.di.ens.fr/~mallat/papiers/math_multiresolution.pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Multiresolution+approximations+and+wavelet+orthonormal+bases+of+L2%28%27%27%27R%27%27%27%29&amp;rft.jtitle=Trans.+Amer.+Math.+Soc.&amp;rft.issue=1&amp;rft.aulast=Mallat&amp;rft.aufirst=St%C3%A9phane&amp;rft.date=1989&amp;rft.volume=315&amp;rft.pages=69-87&amp;rft_id=info%3Adoi%2F10.2307%2F2001373&amp;rft_id=http%3A%2F%2Fwww.di.ens.fr%2F~mallat%2Fpapiers%2Fmath_multiresolution.pdf&amp;rfr_id=info%3Asid%2Ffr.wikipedia.org%3AMultir%C3%A9solution"></span></span></span>.</span>
</li>
</ol></div>
</div>
<div class="mw-heading mw-heading2"><h2 id="Bibliographie">Bibliographie</h2></div>
<p><a href="Yves_Meyer" title="Yves Meyer">Yves Meyer</a>, <i><a href="Ondelette" title="Ondelette">Ondelettes</a> et opérateurs</i>, vol. I, Hermann, 1990
</p>
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