View source for Definite kernel

<!--
d0306901.png
$#A+1 = 7 n = 0
$#C+1 = 7 : ~/encyclopedia/old_files/data/D030/D.0300690 Definite kernel
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The kernel  $  K ( P , Q ) $
of a linear integral [[Fredholm-operator(2)|Fredholm operator]] which satisfies the relation

$$ 
\int\limits _ { P } \int\limits _ { Q } K ( P , Q ) \phi ( P) \overline{ {\phi ( Q) }}\;
 d P  d Q  \geq   0 \  ( \leq   0 ) ,
$$

where  $  P , Q $
are points in a Euclidean space,  $  \phi $
is an arbitrary square-integrable function, and  $  \overline \phi \; $
is its complex conjugate function. Depending on the sign of the inequality, the kernel is said to be, respectively, non-negative (non-negative definite) or non-positive (non-positive definite).

Non-negative (non-positive) is sometimes the name given to a kernel which satisfies, in the domain of integration, the inequality  $  K ( P , Q ) \geq  0 $(
$  \leq  0 $).

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Zaanen,   "Linear analysis" , North-Holland  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jörgens,   "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR></table>