On the Moment Functionals with Signed Representing Measures

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