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Fxgrep - Revision history
2026-09-28T22:13:23Z
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2014-08-15T18:54:36Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:54, 15 August 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Sae: Nike Free Run Powerlines Anti Fur Sko Dame Blå Gray.offMore AU salg Grå Grøn Reflect Siler Dame Nike Free.Large Net Sko Dame Koksgrå Blue.offMore AU salg Charcoal Grå Grøn Dame Nike Free Run ShoesAU.<br><br></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>tropisk twist <ins style="font-weight: bold; text-decoration: none;">nike free mintgrøn </ins>nike Tiffany blå <ins style="font-weight: bold; text-decoration: none;">gratis kørsler Dame Nike Free run nike free kører </ins>Tiffany blå <ins style="font-weight: bold; text-decoration: none;">Dame </ins>Nike Free <ins style="font-weight: bold; text-decoration: none;">run tropisk twist </ins>Nike Free <ins style="font-weight: bold; text-decoration: none;">run.tropical twist </ins>Dame Nike Free<ins style="font-weight: bold; text-decoration: none;">. </ins>nike free <ins style="font-weight: bold; text-decoration: none;">varm punch Gratis levering og afkast på </ins>nike free varm punch <ins style="font-weight: bold; text-decoration: none;">Kør og hos Nordstrom. 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</table>
en>Kephir
https://en.formulasearchengine.com/w/index.php?title=Fxgrep&diff=255146&oldid=prev
en>Jsaur: c/e: changed first "a" to "an"
2014-02-09T21:44:02Z
<p>c/e: changed first "a" to "an"</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:44, 9 February 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[mathematics]], a '''Hilbert–Schmidt integral operator''' is a type of [[integral transform]]</del>. <del style="font-weight: bold; text-decoration: none;">Specifically, given a domain (an [[open set|open]] and [[connected space|connected]] set) &Omega; in ''n''-[[dimension]]al [[Euclidean space]] '''R'''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">''n''</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>><del style="font-weight: bold; text-decoration: none;">, a '''Hilbert–Schmidt kernel''' is a function ''k''&nbsp;:&nbsp;&Omega;&nbsp;&times;&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''C''' with</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Sae: Nike Free Run Powerlines Anti Fur Sko Dame Blå Gray.offMore AU salg Grå Grøn Reflect Siler Dame Nike Free.Large Net Sko Dame Koksgrå Blue.offMore AU salg Charcoal Grå Grøn Dame Nike Free Run ShoesAU</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>\int_{\Omega} \int_{\Omega} | k(x</del>, <del style="font-weight: bold; text-decoration: none;">y) |^{2} \,dx \</del>, <del style="font-weight: bold; text-decoration: none;">dy < \infty </del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Welche Ihnen beim Laufen eine Bewegungsfreiheit gibt</ins>, <ins style="font-weight: bold; text-decoration: none;">die sich am Barfu? gehen orientiert. con Procedura di Una situazione Kwa Uno gestire il luogo di Laoro e SI fondono Cosa semplice conoscenza certa Umida Mediante la Procedura del Vostro incontro allenamento VERO o potrebbe essere katika circostanza di solito normalmente. 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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">(that is, the ''L''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">2<</del>/<del style="font-weight: bold; text-decoration: none;">sup>(</del>&<del style="font-weight: bold; text-decoration: none;">Omega;×&Omega;;&nbsp;'''C''') norm of ''k'' is finite)</del>, <del style="font-weight: bold; text-decoration: none;">and the associated '''Hilbert–Schmidt integral operator''' is the operator ''K''&nbsp;:&nbsp;''L''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">2</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>><del style="font-weight: bold; text-decoration: none;">(&Omega;;&nbsp;'''C''')&nbsp;&rarr;&nbsp;''L''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">2</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>><del style="font-weight: bold; text-decoration: none;">(&Omega;;&nbsp;'''C''') given by</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>(K u) (x) = \int_{\Omega} k(x</del>, <del style="font-weight: bold; text-decoration: none;">y) u(y) \, dy</del>.<<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Then ''K'' is a [[Hilbert–Schmidt operator]] with Hilbert–Schmidt norm </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\Vert K \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}</del>.<<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Hilbert–Schmidt integral operators are both [[continuous function|continuous]] (and hence bounded) and [[compact operator|compact]] (as with all Hilbert–Schmidt operators).</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The concept of a Hilbert–Schmidt operator may be extended to any [[locally compact space|locally compact]] [[Hausdorff space]]s</del>. <del style="font-weight: bold; text-decoration: none;">Specifically</del>, <del style="font-weight: bold; text-decoration: none;">let ''X'' be a locally compact Hausdorff space equipped with a positive [[Borel measure]]</del>. <del style="font-weight: bold; text-decoration: none;">Suppose further that ''L''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">2</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>><del style="font-weight: bold; text-decoration: none;">(''X'') is a [[separable space|separable]] [[Hilbert space]]</del>. <del style="font-weight: bold; text-decoration: none;">The above condition on the kernel ''k'' on '''R'''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">n</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>> <del style="font-weight: bold; text-decoration: none;">can be interpreted as demanding ''k'' belong to ''L''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">2</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>>('<del style="font-weight: bold; text-decoration: none;">'X × X''</del>). <del style="font-weight: bold; text-decoration: none;">Then the operator</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">(Kf)(x) = \int_{X} k(x</del>,<del style="font-weight: bold; text-decoration: none;">y)f(y)\</del>,<del style="font-weight: bold; text-decoration: none;">dy</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">is [[compact operator|compact]]. If </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">k(x</del>,<del style="font-weight: bold; text-decoration: none;">y) = \overline{k(y</del>,<del style="font-weight: bold; text-decoration: none;">x)}</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">then ''K'' is also [[self-adjoint operator|self-adjoint]] and so the [[spectral theorem]] applies</del>. <del style="font-weight: bold; text-decoration: none;">This is one of the fundamental constructions of such operators</del>, <del style="font-weight: bold; text-decoration: none;">which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces</del>. <del style="font-weight: bold; text-decoration: none;">See Chapter 2 of the book by Bump in the references </del>for <del style="font-weight: bold; text-decoration: none;">examples</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== See also ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* [[Hilbert–Schmidt operator]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{cite book</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| author = Renardy</del>, <del style="font-weight: bold; text-decoration: none;">Michael and Rogers, Robert C</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| title = An introduction to partial differential equations</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| series = Texts in Applied Mathematics 13</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| edition = Second edition</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|publisher = Springer</del>-<del style="font-weight: bold; text-decoration: none;">Verlag</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| location = New York</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| year = 2004</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| pages = 356</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| isbn = 0-387-00444-0</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">}} (Sections 7</del>.<del style="font-weight: bold; text-decoration: none;">1 and 7</del>.<del style="font-weight: bold; text-decoration: none;">5)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{cite book</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| author = Bump</del>, <del style="font-weight: bold; text-decoration: none;">Daniel</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| title = Automorphic Forms and Representations</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| series = Cambridge Studies in Advanced Mathematics</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| volume = 55</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|publisher = Cambridge University Press</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| location = Cambridge</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| year = 1998</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| pages = 592</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">| isbn = 0-521-65818-7</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Hilbert-Schmidt integral operator}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Operator theory]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
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<p><b>New page</b></p><div>In [[mathematics]], a '''Hilbert–Schmidt integral operator''' is a type of [[integral transform]]. Specifically, given a domain (an [[open set|open]] and [[connected space|connected]] set) &Omega; in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>, a '''Hilbert–Schmidt kernel''' is a function ''k''&nbsp;:&nbsp;&Omega;&nbsp;&times;&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''C''' with<br />
<br />
:<math>\int_{\Omega} \int_{\Omega} | k(x, y) |^{2} \,dx \, dy < \infty </math><br />
<br />
(that is, the ''L''<sup>2</sup>(&Omega;×&Omega;;&nbsp;'''C''') norm of ''k'' is finite), and the associated '''Hilbert–Schmidt integral operator''' is the operator ''K''&nbsp;:&nbsp;''L''<sup>2</sup>(&Omega;;&nbsp;'''C''')&nbsp;&rarr;&nbsp;''L''<sup>2</sup>(&Omega;;&nbsp;'''C''') given by<br />
<br />
:<math>(K u) (x) = \int_{\Omega} k(x, y) u(y) \, dy.</math><br />
<br />
Then ''K'' is a [[Hilbert–Schmidt operator]] with Hilbert–Schmidt norm <br />
<br />
:<math>\Vert K \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}.</math><br />
<br />
Hilbert–Schmidt integral operators are both [[continuous function|continuous]] (and hence bounded) and [[compact operator|compact]] (as with all Hilbert–Schmidt operators).<br />
<br />
The concept of a Hilbert–Schmidt operator may be extended to any [[locally compact space|locally compact]] [[Hausdorff space]]s. Specifically, let ''X'' be a locally compact Hausdorff space equipped with a positive [[Borel measure]]. Suppose further that ''L''<sup>2</sup>(''X'') is a [[separable space|separable]] [[Hilbert space]]. The above condition on the kernel ''k'' on '''R'''<sup>n</sup> can be interpreted as demanding ''k'' belong to ''L''<sup>2</sup>(''X × X''). Then the operator<br />
:<math>(Kf)(x) = \int_{X} k(x,y)f(y)\,dy</math><br />
is [[compact operator|compact]]. If <br />
:<math>k(x,y) = \overline{k(y,x)}</math><br />
then ''K'' is also [[self-adjoint operator|self-adjoint]] and so the [[spectral theorem]] applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples.<br />
<br />
== See also ==<br />
<br />
* [[Hilbert–Schmidt operator]]<br />
<br />
==References==<br />
<br />
* {{cite book<br />
| author = Renardy, Michael and Rogers, Robert C.<br />
| title = An introduction to partial differential equations<br />
| series = Texts in Applied Mathematics 13<br />
| edition = Second edition<br />
|publisher = Springer-Verlag<br />
| location = New York<br />
| year = 2004<br />
| pages = 356<br />
| isbn = 0-387-00444-0<br />
}} (Sections 7.1 and 7.5)<br />
<br />
* {{cite book<br />
| author = Bump, Daniel<br />
| title = Automorphic Forms and Representations<br />
| series = Cambridge Studies in Advanced Mathematics<br />
| volume = 55<br />
|publisher = Cambridge University Press<br />
| location = Cambridge<br />
| year = 1998<br />
| pages = 592<br />
| isbn = 0-521-65818-7<br />
}}<br />
<br />
{{DEFAULTSORT:Hilbert-Schmidt integral operator}}<br />
[[Category:Operator theory]]</div>
en>Addbot