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Difference between revisions of "Degenerate kernel"

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A kernel of a linear Fredholm integral operator (cf. [[Fredholm-operator(2)|Fredholm operator]]) of the form
 
A kernel of a linear Fredholm integral operator (cf. [[Fredholm-operator(2)|Fredholm operator]]) of the form
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030820/d0308201.png" /></td> </tr></table>
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\sum_{i=1}^N \phi_i(P) \psi_i(Q)
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$$
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030820/d0308202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030820/d0308203.png" /> are points in Euclidean spaces.
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where $P$ and $Q$ are points in Euclidean spaces.
  
  
  
 
====Comments====
 
====Comments====
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A linear Fredholm integral operator with a square-integrable kernel, considered as an operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030820/d0308204.png" />, has a finite-dimensional range if and only if its kernel is degenerate.
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A linear Fredholm integral operator with a square-integrable kernel, considered as an operator on $L_2$, has a finite-dimensional range if and only if its kernel is degenerate.
  
 
Linear Fredholm integral equations of the second kind can be reduced to a system of linear algebraic equations if the kernel is degenerate.
 
Linear Fredholm integral equations of the second kind can be reduced to a system of linear algebraic equations if the kernel is degenerate.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley (Interscience)  (1973)  pp. Sect. 2.5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1958)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley (Interscience)  (1973)  pp. Sect. 2.5</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1958)  (Translated from Russian)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 17:21, 18 October 2014

A kernel of a linear Fredholm integral operator (cf. Fredholm operator) of the form $$ \sum_{i=1}^N \phi_i(P) \psi_i(Q) $$ where $P$ and $Q$ are points in Euclidean spaces.


Comments

A linear Fredholm integral operator with a square-integrable kernel, considered as an operator on $L_2$, has a finite-dimensional range if and only if its kernel is degenerate.

Linear Fredholm integral equations of the second kind can be reduced to a system of linear algebraic equations if the kernel is degenerate.

References

[a1] H. Hochstadt, "Integral equations" , Wiley (Interscience) (1973) pp. Sect. 2.5
[a2] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a3] L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)
[a4] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1958) (Translated from Russian)
How to Cite This Entry:
Degenerate kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_kernel&oldid=12819
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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