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Difference between revisions of "Definite kernel"

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The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306901.png" /> of a linear integral [[Fredholm-operator(2)|Fredholm operator]] which satisfies the relation
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<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/d/d030/d030690/d0306902.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306903.png" /> are points in a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306904.png" /> is an arbitrary square-integrable function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306905.png" /> 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).
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The kernel $  K ( P , Q ) $
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of a linear integral [[Fredholm-operator(2)|Fredholm operator]] which satisfies the relation
  
Non-negative (non-positive) is sometimes the name given to a kernel which satisfies, in the domain of integration, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306906.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030690/d0306907.png" />).
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$$
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\int\limits _ { P } \int\limits _ { Q } K ( P , Q ) \phi ( P) \overline{ {\phi ( Q) }}\;
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d P  d Q  \geq  0 ( \leq  0 ) ,
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$$
  
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where  $  P , Q $
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are points in a Euclidean space,  $  \phi $
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is an arbitrary square-integrable function, and  $  \overline \phi \; $
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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).
  
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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 $(
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$  \leq  0 $).
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 17:32, 5 June 2020


The kernel $ K ( P , Q ) $ of a linear integral 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

[a1] A.C. Zaanen, "Linear analysis" , North-Holland (1956)
[a2] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
How to Cite This Entry:
Definite kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Definite_kernel&oldid=19084
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article