Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-05T11:51:16.867Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Widom, H.Asymptotic behaviour of the eigenvalues of certain integral equations. II. Arch. Rational Mech. Anal. 17 (1964), 215–229.CrossRefGoogle Scholar
(2)Greenhill, A. G.Applications of elliptic functions (Dover, 1959).Google Scholar
(3)Neville, E. H.Jacobian elliptic functions (Oxford, 1951).Google Scholar
(4)Phillips, E. G.Some topics in complex analysis (Pergamon, 1966).Google Scholar
(5)Pietsch, A.Nuclear locally convex spaces. Ergebnisse der mathematik, vol. 66 (Springer 1972).Google Scholar
(6)Schatten, R. Norm ideals of completely continuous operators. Ergebnisse der Mathematik, 27 (Springer, 1960).Google Scholar
(7)Duren, P. L.Theory of Hp-spaces (Academic Press, 1970).Google Scholar
(8)Riesz, F. and Sz-Nagy, B., trans. Boron, L. F.Functional analysis (Ungar, 1955).Google Scholar
(9)Abramowitz, M. and Stegun, I. (eds). Handbook of mathematical functions (Dover, 1965).Google Scholar
(10)Smithies, F.Integral equations (Cambridge, 1958).Google Scholar