@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-P4VMDJPB-V
  skos:prefLabel "espace fonctionnel"@fr, "function space"@en ;
  a skos:Concept ;
  skos:narrower psr:-TXTD8V7M-1 .

psr:-X5ZLLBMK-J
  skos:prefLabel "spherical harmonic"@en, "harmonique sphérique"@fr ;
  a skos:Concept ;
  skos:broader psr:-TXTD8V7M-1 .

psr:-B7HJTVNW-N
  skos:prefLabel "fonction de Wannier"@fr, "Wannier function"@en ;
  a skos:Concept ;
  skos:broader psr:-TXTD8V7M-1 .

psr: a skos:ConceptScheme .
psr:-TXTD8V7M-1
  skos:narrower psr:-X5ZLLBMK-J, psr:-B7HJTVNW-N ;
  skos:definition """In mathematics, <b>orthogonal functions</b> belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:
<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 \\\\langle f,g\\
angle =\\\\int {\\\\overline {f(x)}}g(x)\\\\,dx.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mi>g</mi>
<br/>        <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<br/>        <mo>=</mo>
<br/>        <mo>∫<!-- ∫ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mover>
<br/>            <mrow>
<br/>              <mi>f</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mo accent="false">¯<!-- ¯ --></mo>
<br/>          </mover>
<br/>        </mrow>
<br/>        <mi>g</mi>
<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>x</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\langle f,g\\
angle =\\\\int {\\\\overline {f(x)}}g(x)\\\\,dx.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba2c50db5bca351c8f20dd0d4fe13f4ad5d792c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:23.284ex; height:5.676ex;" alt="{\\\\displaystyle \\\\langle f,g\\
angle =\\\\int {\\\\overline {f(x)}}g(x)\\\\,dx.}"></span></dd></dl>
<br/>The functions <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 f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> and <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 g}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>g</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle g}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"></span> are orthogonal when this integral is zero, i.e. <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 \\\\langle f,\\\\,g\\
angle =0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>g</mi>
<br/>        <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\langle f,\\\\,g\\
angle =0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165997d42a653716f9f90a62236435c259f84b89" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.886ex; height:2.843ex;" alt="{\\\\displaystyle \\\\langle f,\\\\,g\\
angle =0}"></span> whenever <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 f\\
eq g}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>≠<!-- ≠ --></mo>
<br/>        <mi>g</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f\\
eq g}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06f1315a1fb0ac926953e122a609d2dcd5203eeb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.676ex;" alt="{\\\\displaystyle f\\
eq g}"></span>. As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.  Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Orthogonal_functions">https://en.wikipedia.org/wiki/Orthogonal_functions</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-P4VMDJPB-V ;
  skos:prefLabel "orthogonal function"@en, "fonction orthogonale"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Orthogonal_functions> .