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Asymptotic estimates of the eigenvalues of certain positive Fredholm operators

Published online by Cambridge University Press:  24 October 2008

G. Little
G. Little
Affiliation:
University of Manchester
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Extract

1. Introduction. Suppose that K is a continuous function on the square Q = [ – 1, 1] x [– 1,1] satisfying , for – 1 ≤ s, t ≤ 1; then the Fredholm operator T on L2(-1,1)

is compact and symmetric. Suppose also that T is a positive operator, i.e.

then there is an eigenfunction expansion

where (λn) is a sequence of non-negative real numbers which decreases to 0 and (φn) is an orthonormal sequence in L2( – 1,1). In this paper we shall find asymptotic estimates for λn when K takes certain specific analytic forms. In all cases K will be real-valued on Q and analytic in a neighbourhood of Q in complex 2-space; for example

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 2 , March 1982 , pp. 267 - 284
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

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