The smallest eigenvalue of Hankel matrices

TY - JOUR

T1 - The smallest eigenvalue of Hankel matrices

AU - Berg, Christian

AU - Szwarc, Ryszard

PY - 2011/8

Y1 - 2011/8

N2 - Let HN=(sn+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 λN of HN. It is proven that λN has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λN can be arbitrarily slow or arbitrarily fast in a sense made precise below. In the indeterminate case, where λN is known to be bounded below by a strictly positive constant, we prove that the limit of the nth smallest eigenvalue of HN for N→∞ tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

AB - Let HN=(sn+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 λN of HN. It is proven that λN has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λN can be arbitrarily slow or arbitrarily fast in a sense made precise below. In the indeterminate case, where λN is known to be bounded below by a strictly positive constant, we prove that the limit of the nth smallest eigenvalue of HN for N→∞ tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

U2 - 10.1007/s00365-010-9109-4

DO - 10.1007/s00365-010-9109-4

M3 - Journal article

SN - 0176-4276

VL - 34

SP - 107

EP - 133

JO - Constructive Approximation

JF - Constructive Approximation

IS - 1

ER -