Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T01:01:05.245Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigen-values of Lipschitz kernels

Published online by Cambridge University Press:  24 October 2008

J. B. Reade
J. B. Reade
Affiliation:
University of Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic estimates are obtained for the eigen-values of symmetric kernels which satisfy a Lipschitz condition of order α, where 0 < α ≤ 1, on a bounded region. For 0 < α < 1, the eigen-values are shown to be O(1/nα+½) for general kernels, and O(1/nα+1) for positive definite kernels. For α = 1, the eigen-values are shown to be O(1/n+3/2) in general, but only O(1/n2) when the kernel is positive definite. Examples are given to show these estimates are best possible for powers of n.

The estimates for general kernels are known. In 1931, Hille and Tamarkin used infinite determinants to obtain weaker estimates in the case 0 < α < 1, but the same estimates as ours for α = 1. In 1937, Smithies sharpened the estimates for 0 < α < 1 to the ones we give by a variant of the method used by Weyl in 1912 for smooth kernels. Cochran obtained the same estimates in 1972 using Fredholm determinants. Our proof uses Weyl's method in its original form for 0 < α < 1, and in Smithies' variant for α = 1, so in all cases simplifies previous proofs.

The estimates for positive definite kernels are, as far as we know, new. The proofs use the method of our recent paper on smooth positive definite kernels.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 135 - 140
Copyright
Copyright © Cambridge Philosophical Society 1983

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)Burkill, J. C.The Lebesgue integral (Cambridge, 1958).Google Scholar
(2)Cochran, J. A.Analysis of linear integral equations (Bell Telephone Laboratories, 1972.)Google Scholar
(3)Hiille, E. and Tamarkin, J. E.On the characteristic values of linear integral equations. Acta Math. 57 (1931), 1–76.CrossRefGoogle Scholar
(4)Reade, J. B.Eigen-values of positive definite kernels. S.I.A.M. Journal in Analysis. (To appear.)Google Scholar
(5)Riesz, F. and Nagy, B. S.Functional analysis (Budapest and Szeged, 1952.)Google Scholar
(6)Rudin, W.Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10 (1959), 855–859.CrossRefGoogle Scholar
(7)Smithies, F.The eigen-values and singular values of integral equations. Proc. London Math. Soc. (2) 43 (1937), 255–279.Google Scholar
(8)Weyl, H.Das Asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71 (1912), 441–479.CrossRefGoogle Scholar
(9)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th ed. (Cambridge, 1927.)Google Scholar
(10)Zygmund, A.Trigonometrical series (Wilno, 1935.)Google Scholar