Mutual kernels
From Encyclopedia of Mathematics
reciprocal kernels
Two functions and
of real variables
(or, in general, of points
and
of a Euclidean space), defined on the square
and satisfying the condition
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If a kernel reciprocal with
exists, then
is the resolvent kernel of the integral Fredholm equation
![]() | (*) |
Comments
Indeed, when and
are reciprocal kernels, the solution of equation (*) above is given by
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Consider the Fredholm equation
![]() | (a1) |
and the iterated kernels ,
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Form the Neumann series
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If is continuous on
, this series is uniformly convergent for
small. Then
satisfies
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and
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solves (a1).
The terminology "mutual kernels" and "reciprocal kernels" is rarely used.
References
[a1] | V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) |
[a2] | P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian) |
[a3] | B.L. Moiseiwitsch, "Integral equations" , Longman (1977) |
How to Cite This Entry:
Mutual kernels. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mutual_kernels&oldid=47943
Mutual kernels. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mutual_kernels&oldid=47943
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article