View source for Positive-definite kernel

A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739001.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739003.png" /> is any set, which satisfies the condition

<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/p/p073/p073900/p0739004.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739007.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739008.png" />. The measurable positive-definite kernels on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p0739009.png" /> correspond to the positive integral operators (cf. [[Integral operator|Integral operator]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p07390010.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p07390011.png" /> on a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p07390012.png" /> to be positive definite it is necessary and sufficient that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p07390013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073900/p07390014.png" /> 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>