The smallest eigenvalue of Hankel matrices

TL;DR

This paper investigates the asymptotic behavior of the smallest eigenvalue of Hankel matrices derived from moment sequences, revealing exponential decay in the determinate case and contrasting behavior in the indeterminate case.

Contribution

It provides a comprehensive analysis of the decay rates of the smallest eigenvalue for Hankel matrices associated with moment problems, including special cases like Stieltjes-Wigert polynomials.

Findings

Exponential decay of the smallest eigenvalue for measures with compact support.

Arbitrary slow or fast decay in determinate moment problems.

Rapid divergence of eigenvalues in the indeterminate case.

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.