@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-TL90LWRL-H
  skos:broader psr:-SKTRS1V0-R, psr:-XJ7K95G7-L, psr:-SKGJ9CKK-N, psr:-LFQDHGDQ-7 ;
  skos:definition """En analyse mathématique, le <b>problème des moments</b> est un problème inverse consistant à reconstruire une mesure réelle sur un intervalle donné à partir de ses moments. Plus concrètement, étant donnés un intervalle réel <span class="texhtml mvar" style="font-style:italic;">I</span> et une suite <span class="texhtml">(<i>m<sub>n</sub></i>)</span> de réels, on peut se demander s'il existe sur <span class="texhtml mvar" style="font-style:italic;">I</span> une mesure de Borel (donc positive) <span class="texhtml mvar" style="font-style:italic;">μ</span> telle que pour tout entier naturel <span class="texhtml mvar" style="font-style:italic;">n</span>,
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle m_{n}=\\\\int x^{n}~\\\\mathrm {d} \\\\mu (x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>m</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mo>∫<!-- ∫ --></mo>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">d</mi>
<br/>        </mrow>
<br/>        <mi>μ<!-- μ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle m_{n}=\\\\int x^{n}~\\\\mathrm {d} \\\\mu (x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749b1076c3a75d0f018f382f351145080dfa90b6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:17.9ex; height:5.676ex;" alt="{\\\\displaystyle m_{n}=\\\\int x^{n}~\\\\mathrm {d} \\\\mu (x)}"></span></dd></dl>
<br/>et, le cas échéant, si une telle mesure est unique.
<br/>Si cette mesure existe, elle représente alors la loi de probabilité d’une variable aléatoire réelle dont les moments sont les nombres <span class="texhtml mvar" style="font-style:italic;">m<sub>n</sub></span>.
<br/>On peut noter plusieurs variantes du «&nbsp;problème des moments&nbsp;» selon la forme de l’intervalle&nbsp;:
<br/>
<br/><ul><li>de Hamburger si l'intervalle <span class="texhtml mvar" style="font-style:italic;">I</span> est <span class="mwe-math-element"><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 {R} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\\\\mathbb {R} "></span> tout entier&nbsp;;</li>
<br/><li>de Stieltjes s'il est égal à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle [0,+\\\\infty [}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">[</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mo>+</mo>
<br/>        <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>        <mo stretchy="false">[</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle [0,+\\\\infty [}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207d226525287a9b2ebb3ba52c61454a0df207b2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="[0,+\\\\infty ["></span>&nbsp;;</li>
<br/><li>de Hausdorff si <span class="texhtml mvar" style="font-style:italic;">I</span> est un segment <span class="texhtml">[<i>a</i>,<i>b</i>]</span>.</li> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Probl%C3%A8me_des_moments">https://fr.wikipedia.org/wiki/Probl%C3%A8me_des_moments</a>)"""@fr, """In mathematics, a <b>moment problem</b> arises as the result of trying to invert the mapping that takes a measure <i>μ</i> to the sequence of moments
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }x^{n}\\\\,d\\\\mu (x)\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>m</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>μ<!-- μ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }x^{n}\\\\,d\\\\mu (x)\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539a3c61b39139954d04f706ae0e40231efa8d52" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:20.917ex; height:6.009ex;" alt="{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }x^{n}\\\\,d\\\\mu (x)\\\\,.}"></span></dd></dl>
<br/>More generally, one may consider
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }M_{n}(x)\\\\,d\\\\mu (x)\\\\,.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>m</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>μ<!-- μ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }M_{n}(x)\\\\,d\\\\mu (x)\\\\,.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27e34d419b0dbdd7232f66db9b581758b7a485e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:24.98ex; height:6.009ex;" alt="{\\\\displaystyle m_{n}=\\\\int _{-\\\\infty }^{\\\\infty }M_{n}(x)\\\\,d\\\\mu (x)\\\\,.}"></span></dd></dl>
<br/>for an arbitrary sequence of functions <i>M</i><sub><i>n</i></sub>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Moment_problem">https://en.wikipedia.org/wiki/Moment_problem</a>)"""@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Probl%C3%A8me_des_moments>, <https://en.wikipedia.org/wiki/Moment_problem> ;
  dc:created "2023-08-17"^^xsd:date ;
  skos:inScheme psr: ;
  skos:related psr:-XM9K9XL8-S ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:prefLabel "problème des moments"@fr, "moment problem"@en ;
  a skos:Concept .

psr:-XM9K9XL8-S
  skos:prefLabel "measure"@en, "mesure"@fr ;
  a skos:Concept ;
  skos:related psr:-TL90LWRL-H .

psr:-SKGJ9CKK-N
  skos:prefLabel "géométrie algébrique"@fr, "algebraic geometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-TL90LWRL-H .

psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-TL90LWRL-H .

psr:-LFQDHGDQ-7
  skos:prefLabel "moment"@fr, "moment"@en ;
  a skos:Concept ;
  skos:narrower psr:-TL90LWRL-H .

psr: a skos:ConceptScheme .
psr:-XJ7K95G7-L
  skos:prefLabel "optimization"@en, "optimisation"@fr ;
  a skos:Concept ;
  skos:narrower psr:-TL90LWRL-H .