The complex moment problem: determinacy and extendibility

TL;DR

This paper investigates the uniqueness and extendibility of complex moment sequences, establishing conditions under which positive definite extensions are unique and exploring the structure of representing measures.

Contribution

It proves the uniqueness of positive definite extensions for determinate complex moment sequences with atomless measures at zero and explores related open problems and special cases.

Findings

Extension is unique if the sequence is determinate and has no atom at zero.

Partial solutions are provided for measures supported on algebraic curves.

Examples illustrate the theoretical results and special cases.

Abstract

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb C$ as well as complex moment sequences which are constant on a family of parallel "Diophantine lines". All this is supported by a bunch of illustrative examples.