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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

REFERENCES

[1]Boas, R. P. Jr. Integrability Theorems for Trigonometric Transforms (Springer-Verlag, 1967).CrossRefGoogle Scholar
[2]Chang, S.-H.. A generalization of a theorem of Hille and Tamarkin with applications. Proc. London Math. Soc. (3) 2 (1952), 22–29.CrossRefGoogle Scholar
[3]Cochran, J. A.. The Analysis of Linear Integral Equations (McGraw-Hill, 1972).Google Scholar
[4]Cochran, J. A.. Summability of singular values of ℒ2 kernels–Analogies with Fourier series. Enseign. Math. (2), 22 (1976), 141–157.Google Scholar
[5]Ha, C.-W.. Eigenvalues of differentiable positive definite kernels. SIAM J. Math. Anal. 17 (1986), 415–419.CrossRefGoogle Scholar
[6]Kadota, T. T.. Term-by-term differentiability of Mercer's expansion. Proc. Amer. Math. Soc. 18 (1967), 69–72.Google Scholar
[7]Lorentz, G. G.. Fourier-Koeffizienten und Funktionenklassen. Math. Z. 51 (1948), 135–149.CrossRefGoogle Scholar
[8]Marcus, M.. Basic Theorems in Matrix Theory. NBS Appl. Math. Series 57 (U.S. Dept. of Commerce, 1964).Google Scholar
[9]Oehring, C.. (Private communication.)Google Scholar
[10]Oehring, C.. Smooth kernels and Fourier coefficients. (To appear.)Google Scholar
[11]Reade, J. B.. Eigenvalues of Lipschitz kernels. Math. Proc. Cambridge Philos. Soc. 93 (1983), 135–140.CrossRefGoogle Scholar
[12]Reade, J. B.. Eigenvalues of positive definite kernels. SIAM J. Math. Anal. 14 (1983), 152–157.CrossRefGoogle Scholar
[13]Reade, J. B.. Eigenvalues of positive definite kernels, II. SIAM J. Math. Anal. 15 (1984), 137–142.CrossRefGoogle Scholar
[14]Reade, J. B.. Positive definite C p kernels. SIAM J. Math. Anal. 17 (1986), 420–421.CrossRefGoogle Scholar
[15]Riesz, F. and Sz.-Nagy, B.. Functional Analysis, transl. Boron, L. F. (Unger, 1955).Google Scholar
[16]Schumaker, L. L.. Spline Functions: Basic Theory (John Wiley, 1981).Google Scholar