On the Moment Functionals with Signed Representing Measures

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 $\dR^d$.