Devinatz's moment problem: a description of all solutions

TL;DR

This paper provides a new proof and a complete parameterization of all solutions to Devinatz's moment problem, which involves finding measures in a strip that match given moment sequences, using an operator-theoretic approach.

Contribution

It introduces a novel proof of the Devinatz solvability criterion and fully characterizes all solutions via an abstract operator framework.

Findings

Established a new proof of the Devinatz solvability criterion.

Derived a parameterization of all solutions to the moment problem.

Applied an operator approach leveraging results from Godi, Lucenko, and Shtraus.

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.