On the Moment Functionals with Signed Representing Measures

TL;DR

This paper characterizes when linear functionals on finitely generated real algebras can be represented as integrals with respect to signed measures supported on specific sets, with applications to polynomial algebras in multiple variables.

Contribution

It provides a complete necessary and sufficient characterization of sets and algebras for signed measure representation of linear functionals.

Findings

Characterization of sets K and algebras A for signed measure representation

Application of results to polynomial algebras in multiple variables

Conditions for when all linear functionals are representable as signed measures

Abstract

Suppose that $\sA$ is a finitely generated commutative unital real algebra and $K$ is a closed subset of the set $\hat{A}$ of characters of $\sA$. We study the following problem: When is {\it each} linear functional $L:{\sA} \to \dR$ an integral with respect to some signed Radon measure on $\hat{\sA}$ supported by the set $K$? A complete characterization of the sets $K$ and algebras $\sA$ by necessary and sufficient conditions is given. The result is applied to the polynomial algebra $\dR[x_1,\dots,x_d]$ and subsets $K$ of $\dR^d$.