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The Smallest Eigenvalue of Hankel Matrices

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Abstract

Let ℋ N =(s n+m),0≤n,m≤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 discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

    Christian Berg

  2. Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Ryszard Szwarc

  3. Institute of Mathematics and Computer Science, University of Opole, ul. Oleska 48, 45-052, Opole, Poland

    Ryszard Szwarc

Authors
  1. Christian Berg
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  2. Ryszard Szwarc
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Correspondence to Christian Berg.

Additional information

Communicated by Doron S. Lubinsky.

The 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 272-07-0321 from the Danish Research Council for Nature and the Universe.

The second author was supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285.

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Berg, C., Szwarc, R. The Smallest Eigenvalue of Hankel Matrices. Constr Approx 34, 107–133 (2011). https://doi.org/10.1007/s00365-010-9109-4

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  • DOI: https://doi.org/10.1007/s00365-010-9109-4

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