Difference between revisions of "Definite kernel"
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| + | $#C+1 = 7 : ~/encyclopedia/old_files/data/D030/D.0300690 Definite kernel | ||
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| − | + | 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==== | ====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=46606
Definite kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Definite_kernel&oldid=46606
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article