Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

TL;DR

This paper explores how quasi-analytic function theory is central to understanding when measures are uniquely determined by their moments in both finite and infinite-dimensional spaces, highlighting key criteria and properties.

Contribution

It demonstrates the role of quasi-analytic criteria in the determinacy of the moment problem across finite and infinite dimensions, unifying these perspectives.

Findings

Quasi-analytic criteria are crucial for measure determinacy.

Characterization of quasi-analytic classes linked to log-convex sequences.

Review of known determinacy results from a quasi-analytic perspective.

Abstract

This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasi-analytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.