In this paper we characterise the indeterminate case by the eigenvalues of
the Hankel matrices being bounded below by a strictly positive constant. An
explicit lower bound is given in terms of the orthonormal polynomials and we
find expresions for this lower bound in a number of indeterminate moment
problems.Comment: 14 pages, 1 figur

Berg, Christian

Chen, Yang

Ismail, Mourad E. H.

English

arXiv

In this paper we characterize the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find expressions for this lower bound in a number of indeterminate moment problems. 1 Introduction  Let ff be a positive measure on R with infinite support and finite moments of all orders s n = s n (ff) =  Z  R  x  n  dff(x): (1.1) With ff we associate the infinite Hankel matrix H1 = fH jk g, H jk = s j+k : (1.2) Let HN be the (N+1)\Theta(N +1) matrix whose entries are H jk ; 0  j; k  N . Since HN is positive definite, then all its eigenvalues are positive. The large N asymptotics of the smallest eigenvalue, denoted as  N , of the Hankel matrix HN has been studied in papers by Szego [11], Widom and Wilf [13], Chen and Lawrence [6]. See also the monograph by Wilf [14]. All the cases considered by these authors are determinate moment problems, and it was shown in..

Small eigenvalues of large Hankel matrices:The indeterminate case

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