Moment problem for symmetric algebras of locally convex spaces

TL;DR

This paper extends the moment problem to symmetric algebras of locally convex spaces, providing a representation of positive linear functionals as integrals over Radon measures supported on specific dual space subsets.

Contribution

It introduces a method to extend locally convex topologies to symmetric algebras and characterizes continuous positive functionals as integrals, generalizing previous results.

Findings

Representation of positive functionals as Radon measures

Extension of topology from V to S(V)

Broader applicability to general lc spaces

Abstract

It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of $\overline{\tau}$-continuous linear functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum S(V)^{2d}) \subseteq [0,\infty)$ (more generally, $L(M) \subseteq [0,\infty)$ for some $2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely determined Radon measures $\mu$ supported by special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and \v Sifrin. It is more general because $V$ can be any lc topological space (not just a separable nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that $L : S(V) \rightarrow \mathbb{R}$ is $\overline{\tau}$-continuous (not just continuous on each homogeneous part of $S(V)$).