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Difference between revisions of "Mercer theorem"

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The bilinear series
 
The bilinear series
  
<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/m/m063/m063440/m0634401.png" /></td> </tr></table>
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$$
 +
\sum _ { m }
 +
\frac{\phi _ {m} ( s) \overline{ {\phi _ {m} ( t) }}\; }{
 +
\lambda _ {m} }
 +
 
 +
$$
  
of a Hermitian positive-definite continuous kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634402.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634403.png" /> (cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]; [[Kernel of an integral operator|Kernel of an integral operator]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634404.png" /> is the closure of a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634405.png" />, converges absolutely and uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634406.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634407.png" />. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634408.png" /> are the characteristic numbers of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634409.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344010.png" /> are the corresponding orthonormalized eigen functions. If a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344011.png" /> satisfies the conditions of Mercer's theorem, then the integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344012.png" />,
+
of a Hermitian positive-definite continuous kernel $  K( s, t) $
 +
on $  D \times D $(
 +
cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]; [[Kernel of an integral operator|Kernel of an integral operator]]), where $  D $
 +
is the closure of a bounded domain in $  \mathbf R  ^ {n} $,  
 +
converges absolutely and uniformly in $  D \times D $
 +
to $  K( s, t) $.  
 +
Here the $  \lambda _ {m} $
 +
are the characteristic numbers of the kernel $  K( s, t) $
 +
and the $  \phi _ {m} ( s) $
 +
are the corresponding orthonormalized eigen functions. If a kernel $  K $
 +
satisfies the conditions of Mercer's theorem, then the integral operator $  T: L _ {2} ( D) \rightarrow L _ {2} ( D) $,
  
<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/m/m063/m063440/m06344013.png" /></td> </tr></table>
+
$$
 +
Tf( s)  = \int\limits _ { D } K( s, t) f( t)  dt  = \
 +
\sum _ { m }
 +
\frac{1}{\lambda _ {m} }
 +
( f, \phi _ {m} ) \phi _ {m}  $$
  
is nuclear (cf. [[Nuclear operator|Nuclear operator]]) and its [[Trace|trace]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344014.png" /> can be calculated by the formula
+
is nuclear (cf. [[Nuclear operator|Nuclear operator]]) and its [[Trace|trace]] $  \sum _ {m} 1/ \lambda _ {m} $
 +
can be calculated by the formula
  
<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/m/m063/m063440/m06344015.png" /></td> </tr></table>
+
$$
 +
\sum _ { m }
 +
\frac{1}{\lambda _ {m} }
 +
  = \
 +
\int\limits _ { D } K( s, s)  ds.
 +
$$
  
 
Mercer's theorem can be generalized to the case of a bounded discontinuous kernel.
 
Mercer's theorem can be generalized to the case of a bounded discontinuous kernel.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Mercer,  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''209'''  (1909)  pp. 415–446</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Mercer,  "Functions of positive and negative type, and their connection with the theory of integral equations"  ''Proc. Roy. Soc. London Ser. A'' , '''83'''  (1908)  pp. 69–70</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Krasnosel'skii,  et al.,  "Integral operators in spaces of summable functions" , Noordhoff  (1976)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Mercer,  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''209'''  (1909)  pp. 415–446</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Mercer,  "Functions of positive and negative type, and their connection with the theory of integral equations"  ''Proc. Roy. Soc. London Ser. A'' , '''83'''  (1908)  pp. 69–70</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Krasnosel'skii,  et al.,  "Integral operators in spaces of summable functions" , Noordhoff  (1976)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Linear analysis" , North-Holland  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Linear analysis" , North-Holland  (1956)</TD></TR></table>

Latest revision as of 08:00, 6 June 2020


The bilinear series

$$ \sum _ { m } \frac{\phi _ {m} ( s) \overline{ {\phi _ {m} ( t) }}\; }{ \lambda _ {m} } $$

of a Hermitian positive-definite continuous kernel $ K( s, t) $ on $ D \times D $( cf. Integral equation with symmetric kernel; Kernel of an integral operator), where $ D $ is the closure of a bounded domain in $ \mathbf R ^ {n} $, converges absolutely and uniformly in $ D \times D $ to $ K( s, t) $. Here the $ \lambda _ {m} $ are the characteristic numbers of the kernel $ K( s, t) $ and the $ \phi _ {m} ( s) $ are the corresponding orthonormalized eigen functions. If a kernel $ K $ satisfies the conditions of Mercer's theorem, then the integral operator $ T: L _ {2} ( D) \rightarrow L _ {2} ( D) $,

$$ Tf( s) = \int\limits _ { D } K( s, t) f( t) dt = \ \sum _ { m } \frac{1}{\lambda _ {m} } ( f, \phi _ {m} ) \phi _ {m} $$

is nuclear (cf. Nuclear operator) and its trace $ \sum _ {m} 1/ \lambda _ {m} $ can be calculated by the formula

$$ \sum _ { m } \frac{1}{\lambda _ {m} } = \ \int\limits _ { D } K( s, s) ds. $$

Mercer's theorem can be generalized to the case of a bounded discontinuous kernel.

The theorem was proved by J. Mercer [1].

References

[1] J. Mercer, Philos. Trans. Roy. Soc. London Ser. A , 209 (1909) pp. 415–446
[2] J. Mercer, "Functions of positive and negative type, and their connection with the theory of integral equations" Proc. Roy. Soc. London Ser. A , 83 (1908) pp. 69–70
[3] I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian)
[4] F.G. Tricomi, "Integral equations" , Interscience (1957)
[5] M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian)

Comments

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a2] A.C. Zaanen, "Linear analysis" , North-Holland (1956)
How to Cite This Entry:
Mercer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mercer_theorem&oldid=47822
This article was adapted from an original article by V.B. Korotkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article