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Journal of Operator Theory

Volume 68, Issue 2, Fall 2012  pp. 515-541.

Devinatz's moment problem: a description of all solutions

Authors:  Sergey M. Zagorodnyuk
Author institution: School of Mathematics and Mechanics, Karazin Kharkiv National University, Kharkiv, 61022, Ukraine

Summary:  In this paper we study Devinatz's moment problem: to find a non-negative Borel measure $\mu$ in a strip $\Pi = \{ (x,\varphi):\ x\in \mathbb{R},\ -\pi\leqslant \varphi < \pi \},$ such that $\int\limits_\Pi x^m \mathrm{e}^{\mathrm{i}n\varphi} \mathrm d\mu = s_{m,n}$, $m\in \mathbb{Z}_+$, $n\in \mathbb{Z}$, where $\{ s_{m,n} \}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$ is a given sequence of complex numbers. We derive a solvability criterion for this moment problem. We obtain a parametrization of all solutions of Devinatz's moment problem. We use an abstract operator approach and results of Godič, Lucenko and Shtraus.

Keywords:  Moment problem, measure, generalized resolvent

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