Devinatz's moment problem: a description of all solutions

Abstract

In this paper we study Devinatz's moment problem: to find a non-negative Borel measure $\mu$ in a strip $\Pi = \{(x,\phi):\ x\in \mathbb{R},\ -\pi\leq \phi < \pi\},$ such that $\int_\Pi x^m e^{in\phi} d\mu = s_{m,n}$, $m\in \mathbb{Z}_+$, $n\in \mathbb{Z}$, where $\{s_{m,n}\}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$ is a given sequence of complex numbers. We present a new proof of the Devinatz solvability criterion for this moment problem. We obtained a parameterization of all solutions of Devinatz's moment problem. We used an abstract operator approach and results of Godi\v{c}, Lucenko and Shtraus.