Nevanlinna matrices for the strong Hamburger moment problem
type: Item
instance of: scholarly article
title: Nevanlinna matrices for the strong Hamburger moment problem
zbMATH Open document ID: 1038.30014
author: Olav Njåstad
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
review text: Let \(\{c_n\}^\infty_{n=-\infty}\) be a doubly infinite sequence of real numbers. The strong Hamburger moment problem consists of finding positive measures \(\sigma\) on \(\mathbb R\) such that \(c_n =\int_{-\infty}^\infty t^n\,d\sigma(t)\) for \(n\in\mathbb Z\). 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 \(\varphi\) and all solutions \(\sigma\) of the moment problem, expressed by \(\int_{-\infty}^\infty(z-t)^{-1}\,d\sigma(t)= [\alpha(z)\varphi(z) -\gamma(z)]/[\beta(z)\varphi(z) -\delta(z)]\). The functions \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are holomorphic in the complex plane outside the origin. The purpose of this paper is to study growth properties of these functions at the origin and at infinity, partly analogous to those at infinity of corresponding entire functions connected with the classical Hamburger moment problem. A crucial role is here played by a Riesz criterion for indeterminacy. The theory of orthogonal Laurent polynomials and their use in the study of strong moment problems are sketched, including the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger problem.
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 2060089
zbMATH Keywords: indeterminate strong Hamburger moment problems
zbMATH Keywords: Nevanlinna parametrization
zbMATH Keywords: Nevanlinna matrices
zbMATH Keywords: Pick functions
zbMATH Keywords: Riesz criterion for indeterminacy
zbMATH Keywords: orthogonal Laurent polynomials
zbMATH Keywords: growth estimates
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069963
OpenAlex ID: W2081828807
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DOI: 10.1216/RMJM/1181069963
description: scientific article; zbMATH DE number 2060089