Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions
type: Item
instance of: scholarly article
title: Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions
zbMATH Open document ID: 0837.58018
DOI: 10.1007/BF00994647
author: Yoshimasa Nakamura
author: Yuji Kodama
published in: Acta Applicandae Mathematicae
publication date: 1995-10-22T00:00:00Z
review text: A restricted version of the moment problem of Hamburger [\textit{N. I. Akhiezer} and \textit{M. Krein}, `Some questions in the theory of moments' (1962; Zbl 0117.32702)] is considered. The problem is to show how to construct a parametric Stieltjes measure for a given set of moments depending on parameters. It is shown that if a dynamics of moments is governed by a certain integrable dynamical system (Kac-van Moerbeke system), then the Stieltjes measure can be constructed explicitly by integrating a Moser's nonlinear dynamical system. A necessary and sufficient condition for the existence of solutions of the moment problem was already given by Toeplitz [loc. cit.]. It is shown, that the conditon is essentially equivalent to positivity of tau-functions of the finite and infinite Toda equations.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 790811
zbMATH Keywords: Hamburger moment problem
zbMATH Keywords: Moser's system
zbMATH Keywords: Kac-van Moerbeke system
zbMATH Keywords: tau- functions
MaRDI profile type: Publication
cites work: Completely integrable gradient flows
cites work: Orthogonal polynomials from the viewpoint of scattering theory
cites work: Q3254327
cites work: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices
cites work: Jacobi-Perron algorithms, bi-orthogonal polynomials and inverse scattering problems
cites work: Finitely many mass points on the line under the influence of an exponential potential -- an integrable system
cites work: Geometry of Rational Functions and Nonlinear Integrable Systems
cites work: Q4328360
cites work: Q4061445
full work available at URL: https://doi.org/10.1007/bf00994647
OpenAlex ID: W4236952981
description: scientific article; zbMATH DE number 790811