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Definite kernel

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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