A characterization of moment sequences by extensions

TL;DR

This paper generalizes the fibre theorem for the moment problem to finitely generated real unital algebras and characterizes moment functionals via extensions, providing new proofs for classical results in the complex moment problem.

Contribution

It extends the fibre theorem to broader algebraic settings and offers a new characterization of moment functionals through extensions.

Findings

Generalization of the fibre theorem to finitely generated real unital algebras

Characterization of moment functionals on polynomial algebras via extensions

Reproof of classical results on the complex moment problem

Abstract

The fibre theorem \cite{schm2003} for the moment problem on closed semi-algebraic subsets of $\R^d$ is generalized to finitely generated real unital algebras. A charaterization of moment functionals on the polynomial algebra $\R[x_1,\dots,x_d]$ in terms of extensions is given. These results are applied to reprove basic results on the complex moment problem due to Stochel and Szafraniec \cite{stochelsz} and Bisgaard \cite{bisgaard}.