Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T02:21:29.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Singular numbers of smooth kernels

Published online by Cambridge University Press:  24 October 2008

Charles Oehring
Charles Oehring
Affiliation:
Mathematics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 511 - 514
Copyright
Copyright © Cambridge Philosophical Society 1988

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]Aljančič, S. and Tomič, M.. Über den Stetigkeitsmodul von Fourier–Reihen mit monotonen Koeffizienten. Math. Z. 88 (1965), 274–284.CrossRefGoogle Scholar
[2]Bary, N.. A Treatise on Trigonometric Series, vols. I and II (Pergamon Press, 1964).Google Scholar
[3]Butzer, P. L. and Berens, H.. Semi-groups of Operators and Approximation (Springer-Verlag, 1967).CrossRefGoogle Scholar
[4]Cochran, J. A.. The nuclearity of operators generated by Holder continuous kernels. Proc. Cambridge Philos. Soc. 75 (1974), 351–356.CrossRefGoogle Scholar
[5]Cochran, J. A.. Growth estimates for the singular values of square-integrable kernels. Pacific J. Math. 56 (1975), 51–58.CrossRefGoogle Scholar
[6]Cochran, J. A.. Summability of singular values of L 2 kernels–analogies with Fourier series. Enseign. Math. (2) 22 (1976), 141–157.Google Scholar
[7]Cochran, J. A.. Composite integral operators and nuclearity. Ark. Mat. 15 (1977), 215–222.CrossRefGoogle Scholar
[8]Cochran, J. A.. An extension of an operator inequality for s-numbers. Proc. Amer. Math. Soc. 65 (1977), 44–46.Google Scholar
[9]Dunford, N. and Schwartz, J. T.. Linear Operators, Part II: Spectral Theory (Interscience, 1963).Google Scholar
[10]Halmos, P. R. and Sunder, V.. Bounded Integral Operators on L2 Spaces (Springer-Verlag, 1978).CrossRefGoogle Scholar
[11]Hardy, G. H. and Rogosinski, W.. Notes on Fourier series III: asymptotic formulae for the sums of certain trigonometrical series. Quart. J. Math. 16 (1945), 49–58.CrossRefGoogle Scholar
[12]Oehring, C.. Fourier series and smooth kernels (to be published).Google Scholar
[13]Smithies, F.. The eigen-values and singular values of integral equations. Proc. London Math. Soc. (2) 43 (1937), 255–279.Google Scholar
[14]Smithies, F.. Integral Equations (Cambridge University Press, 1962).Google Scholar
[15]Weidmann, J.. Linear Operators in Hilbert Spaces (Springer-Verlag, 1960).Google Scholar
[16]Weyl, H.. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71 (1912), 441–479.CrossRefGoogle Scholar
[17]Zygmund, A.. Trigonometric Series, vols. I, II (Cambridge University Press, 1959).Google Scholar