The Smallest Eigenvalue of Hankel Matrices

Authors: Christian Berg Ryszard Szwarc

Publish Date: 2010/07/16

Volume: 34, Issue: 1, Pages: 107-133

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Abstract

Let ℋ N =s n+m0≤nm≤N denote the Hankel matrix of moments of a positive measure with moments of any order We study the large N behavior of the smallest eigenvalue λ N of ℋ N It is proven that λ N has exponential decay to zero for any measure with compact support For general determinate moment problems the decay to 0 of λ N can be arbitrarily slow or arbitrarily fast in a sense made precise below In the indeterminate case where λ N is known to be bounded below by a strictly positive constant we prove that the limit of the nth smallest eigenvalue of ℋ N for N→∞ tends rapidly to infinity with n The special case of the Stieltjes–Wigert polynomials is discussedThe present work was initiated while the first author was visiting University of Wrocław under a grant by the HANAP project mentioned under the second author The first author has been supported by grant 272070321 from the Danish Research Council for Nature and the Universe

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