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Differentiable positive definite kernels and Lipschitz continuity

Published online by Cambridge University Press:  24 October 2008

James A. Cochran  and
Mark A. Lukas
James A. Cochran
Affiliation:
Department of Mathematics, Washington State University, Pullman, Washington 99164–2930, U.S.A.
Mark A. Lukas
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Western Australia 6150
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Abstract

Reade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.

The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 361 - 369
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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