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arXiv · 0906.4506

The smallest eigenvalue of Hankel matrices

Christian Berg·Ryszard Szwarc
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Abstract

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

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Christian Berg, Ryszard Szwarc. 2009-06-24. The smallest eigenvalue of Hankel matrices. https://doi.org/10.1007/s00365-010-9109-4

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