Nevanlinna Matrices for the Strong Hamburger Moment Problem

Abstract

Let {c n }∞n=-∞ be a doubly infinite sequence of real numbers. The strong Hamburger moment problem consists of finding positive measures σ on R such that c n = ∫∞ - ∞ t n dσ(t) for n = 0, ±1, ±2,.... The problem is indeterminate if there is more than one solution. For an indeterminate problem there is a one-to-one correspondence between all Pick functions φ and all solutions σ of the moment problem, expressed by ∫∞ - ∞t) -1 dσ(t) = [α(z)φ(z) - γ(z)][β(z)φ(z) - δ(z)] -1 . The functions α,β,γ,δ are holomorphic in the complex plane outside the origin. The purpose of this paper is to study growth properties of these functions a, β,γ,δ, analogous to properties of corresponding entire functions connected with the classical Hamburger moment problem.