Namespaces
Variants
Actions

Difference between revisions of "Positive-definite kernel"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
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
+
<!--
 +
p0739001.png
 +
$#A+1 = 14 n = 0
 +
$#C+1 = 14 : ~/encyclopedia/old_files/data/P073/P.0703900 Positive\AAhdefinite kernel
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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]]].
+
A complex-valued function  $  K $
 +
on $  X \times X $,
 +
where  $  X $
 +
is any set, which satisfies the condition
  
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]]].
+
$$
 +
\sum _ {i,j= 1 } ^ { n }  K( x _ {i} , x _ {j} )
 +
\lambda _ {i} \overline \lambda \; _ {j}  \geq  0,
 +
$$
 +
 
 +
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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) 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 [1].

The theory of positive-definite kernels extends the theory of positive-definite functions (cf. 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 [1].

References

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

Comments

References

[a1] H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)
How to Cite This Entry:
Positive-definite kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_kernel&oldid=48250
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article