On the infinite-dimensional moment problem

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with respect to cylinder measures. Our main results provide such integral representations for $A_{+}$–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra $S(V)$ of a real vector space $V$. As a byproduct, we obtain new approaches to the moment problem on $S(V)$ for a nuclear space $V$ and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra $A$.

Keywords

moment problem, cylinder measure, symmetric algebra, nuclear space, Carleman condition

2010 Mathematics Subject Classification

Primary 44A60. Secondary 28C20, 46G12.

Full Text (PDF format)

Received 10 December 2017

Received revised 4 April 2018

Accepted 27 May 2018

Published 24 May 2022