Cobos, Fernando and Kühn, Thomas (1990) Eigenvalues of Integral-Operators with Positive Definite Kernels Satisfying Integrated Holder. Journal of Approximation Theory, 63 (1). pp. 39-55. ISSN 0021-9045
For a compact metric space X let μ be a finite Borel measure on X. The authors investigate the asymptotic behavior of eigenvalues of integral operators on L2(X, μ). These integral operators are assumed to have a positive definite kernel which satisfies certain conditions of H¨older continuity. For the eigenvalues _n, n 2 N, which are counted according to their algebraic multiplicities and ordered with respect to decreasing absolute values, the main result of this paper consists of estimates _n = O(n−1(_n(X))_) for n ! 1. Here _n(X) represents the entropy numbers of X, and _ is the exponent in the H¨older continuity condition of the kernel. It is shown that in some respect this estimate is optimal. In the special case where X = _ RN is a bounded Borel set, the above estimate yields _n = O(n−_/N−1) for n ! 1. The article concludes with some non-trivial examples of compact metric spaces with regular entropy behavior.
|Uncontrolled Keywords:||Hölder continuity; entropy numbers|
|Subjects:||Sciences > Mathematics > Numerical analysis|
|Deposited On:||07 Jun 2012 08:26|
|Last Modified:||22 Oct 2013 13:48|
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