Nevanlinna matrices for the strong Hamburger moment problem
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.
indeterminate strong Hamburger moment problems
scientific article; zbMATH DE number 2060089