The Smallest Eigenvalue of Hankel Matrices

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 lambda (N) of H-N . It is proven 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 a sense made precise below. In the indeterminate case, where lambda (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 -> a tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.