View source for Hilbert-Schmidt series

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A series of functions

$$ \tag{* }
\sum _ {n = 1 } ^  \infty  

\frac{( f, \phi _ {n} ) }{\lambda _ {n} }
 \phi _ {n} ( x),
$$

where  $  \{ \lambda _ {n} \} $
is the sequence of all eigen values of a symmetric kernel (cf. [[Kernel of an integral operator|Kernel of an integral operator]])  $  K ( x, s) $, 
$  a \leq  x, s \leq  b $, 
$  \{ \phi _ {n} ( x) \} $
is the corresponding sequence of orthonormal eigen functions, while  $  ( f, \phi _ {n} ) $
is the scalar product of an arbitrary square-summable function  $  f $
and the function  $  \phi _ {n} $.

The Hilbert–Schmidt theorem: If the kernel  $  K( x, s) $
is a square-summable function in two variables, then the series (*) converges in the mean to the function

$$ 
\int\limits _ { a } ^ { b }  K ( x, s) f ( s)  ds.
$$

If there exists a constant C such that for all  $  x $
from  $  ( a, b) $
the inequality

$$ 
\int\limits _ { a } ^ { b }  | K ( x, s) |  ^ {2}  ds  \leq   C
$$

is fulfilled, then the Hilbert–Schmidt series converges absolutely and uniformly.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1981)</TD></TR></table>