Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions
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.
scientific article; zbMATH DE number 790811