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Singular numbers of smooth kernels. II

Published online by Cambridge University Press:  24 October 2008

Charles Oehring
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.

Extract

Reade[10] has recently improved Weyl's classical estimate λn = o(n−3/2) for the eigenvalues of a symmetric kernel KC1 by relaxing the Cl hypothesis to the assumptions that KL2[0, 2π]2, that K is absolutely continuous in each variable separately, and that both ∂K/∂s and ∂K/t belong to L2[0, 2π]2. The conclusion of his theorem, that is, of course, stronger than λn = o(n−3/2).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Cochran, J. A.. The nuclearity of operators generated by Hölder continuous kernels. Math. Proc. Cambridge Philos. Soc. 75 (1974), 351356.CrossRefGoogle Scholar
[2]Cochran, J. A.. Summability of singular values of L 2 kernels-analogies with Fourier series. Enseign. Math. (2) 22 (1976), 141157.Google Scholar
[3]DeLeeuw, K., Katznelson, Y. and Kahane, J.. Sur les coefficients de Fourier des fonctions continues. C. R. Acad. Sci. Paris, Sér. A 285 (1977), 10011003.Google Scholar
[4]Dunford, N. and Schwartz, J.. Linear Operators. Part II. Spectral Theory (Interscience, 1963).Google Scholar
[5]Halmos, P. and Sunder, V.. Bounded Integral Operators on L 2 Spaces (Springer-Verlag, 1978).CrossRefGoogle Scholar
[6]Knopp, K.. Theory and Application of Infinite Series (Blackie, 1951).Google Scholar
[7]Oehring, C.. Singular numbers of smooth kernels. Math. Proc. Cambridge Philos. Soc. 103 (1988), 511514.CrossRefGoogle Scholar
[8]Oehring, C.. Asymptotics of singular numbers of smooth kernels via trigonometric transforms. J. Math. Anal. App. (To appear.)Google Scholar
[9]Gohberg, I. and Krein, M.. Introduction to the Theory of Linear Nonselfadjoint Operators (American Mathematical Society, 1969).Google Scholar
[10]Reade, J. B.. Eigenvalues of smooth kernels. Math. Proc. Cambridge Philos. Soc. 95 (1984), 135140.CrossRefGoogle Scholar
[11]Reade, J. B.. On the sharpness of Weyl's estimate for eigenvalues of smooth kernels. SIAM J. Math. Anal. 16 (1985), 548550.CrossRefGoogle Scholar
[12]Smithies, F.. Integral Equations (Cambridge University Press, 1962).Google Scholar
[13]Triebel, H.. Über die Verteilung der Approximationszahlen kompakter Operatoren in Sobolev-Besov Räumen. Invent. Math. 4 (1967), 275293.CrossRefGoogle Scholar
[14]Zygmund, A.. Trigonometric Series, vol. 1 (Cambridge University Press, 1959).Google Scholar
[15]Zygmund, A.. Trigonometric Series, vol. 2 (Cambridge University Press, 1959).Google Scholar