A general fibre theorem for moment problems and some applications

TL;DR

This paper generalizes the fibre theorem for the moment problem to finitely generated real unital algebras, introduces new theorems on the rational multidimensional moment problem, and characterizes moment functionals via extensions.

Contribution

It extends the fibre theorem to a broader algebraic setting and provides new results on the rational multidimensional moment problem and moment functional characterization.

Findings

Generalized fibre theorem for finitely generated algebras

Proved two new theorems on rational multidimensional moment problem

Characterized moment functionals via extensions

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. As an application two new theorems on the rational multidimensional moment problem are proved. Another application is a characterization of moment functionals on the polynomial algebra $\R[x_1,\dots,x_d]$ in terms of extensions. Finally, the fibre theorem and the extension theorem are used to reprove basic results on the complex moment problem due to Stochel and Szafraniec \cite{stochelsz} and Bisgaard \cite{bisgaard}.