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Asymptotic estimates of the eigenvalues of certain positive Fredholm operators. II

Published online by Cambridge University Press:  24 October 2008

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

In a previous paper [1] we gave examples of positive integral operators on L2( − 1, 1) and estimates of their eigenvalues. In theorem 1 we treated operators with kernels of the form Σan sn tn, where (an) is a sequence of non-negative real numbers satisfying an ≃ α2n and 0 < α < 1 (here and throughout the notation an ≃ bn shall mean that an = O(bn) and bn = O(an)). In this paper we prove the more comprehensive Theorem 1 a below; theorem 1 of [1] is just the case b = 0. The term q(2α/(1 + α2)) will be explained immediately after the statement of the result.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 575 - 592
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Little, G.. Asymptotic estimates of the eigenvalues of certain positive Fredholm operators. Math. Proc. Cambridge Philos. Soc. 91 (1982), 267–284.CrossRefGoogle Scholar
[2]Little, G.. Eigenvalues of positive integral operators with certain entire kernels. Math. Proc. Cambridge Philos. Soc. 99 (1986), 535–545.CrossRefGoogle Scholar
[3]Nehari, Z.. Conformal Mapping. International Series in Pure and Applied Mathematics (McGraw-Hill, 1952).Google Scholar
[4]Riesz, F. and Sz-Nagy, B., transl. Boron, L. F.. Functional Analysis (Ungar, 1955).Google Scholar
[5]Titchmarsh, E. C.. The Theory of Functions (Oxford University Press, 1939).Google Scholar