View source for Mercer theorem

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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|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) $,

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

is nuclear (cf. [[Nuclear operator|Nuclear operator]]) and its [[Trace|trace]]  $  \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 [[#References|[1]]].

====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>

====Comments====

====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>