View source for Positive-definite kernel

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A complex-valued function  $  K $
on  $  X \times X $, 
where  $  X $
is any set, which satisfies the condition

$$ 
\sum _ {i,j= 1 } ^ { n }  K( x _ {i} , x _ {j} )
\lambda _ {i} \overline \lambda \; _ {j}  \geq   0,
$$

for any  $  n \in \mathbf N $, 
$  \lambda _ {i} \in \mathbf C $, 
$  x _ {i} \in X $
$  ( i = 1 \dots n) $. 
The measurable positive-definite kernels on a measure space  $  ( X, \mu ) $
correspond to the positive integral operators (cf. [[Integral operator|Integral operator]]) on  $  L _ {2} ( X, \mu ) $; 
in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces [[#References|[1]]].

The theory of positive-definite kernels extends the theory of positive-definite functions (cf. [[Positive-definite function|Positive-definite function]]) on groups: For a function  $  f $
on a group  $  G $
to be positive definite it is necessary and sufficient that the function  $  K( x, y) = f( xy  ^ {-} 1 ) $
on  $  G \times G $
is a positive-definite kernel. In particular, certain results from the theory of positive-definite functions can be extended to positive-definite kernels. For example, Bochner's theorem is that each positive-definite function is the Fourier transform of a positive bounded measure (i.e. an integral linear combination of characters), and this is generalized as follows: Each (generalized) positive-definite kernel has an integral representation by means of so-called elementary positive-definite kernels with respect to a given differential expression [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  M.G. Krein,   "Hermitian positive kernels on homogeneous spaces I"  ''Ukr. Mat. Zh.'' , '''1''' :  4  (1949)  pp. 64–98  (In Russian)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  M.G. Krein,   "Hermitian positive kernels on homogeneous spaces II"  ''Ukr. Mat. Zh.'' , '''2''' :  1  (1950)  pp. 10–59  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press  (1968)</TD></TR></table>