The Moment Problem for Continuous Positive Semidefinite Linear functionals

Abstract

Let $\tau$ be a locally convex topology on the countable dimensional polynomial $\mathbb{R}$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of ${\mathbb{R}}^n$, and let $M:=M_{\{g_1,...g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron.

14 pages

References in corpus (2)

• Closure of the cone of sums of 2d-powers in certain weighted ${\ell }_1$-seminorm topologies
• The K-moment problem for continuous linear functionals

Cited by in corpus (8)

• Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras
• Infinite dimensional moment problem: open questions and applications
• Closure of the cone of sums of 2d-powers in certain weighted ${\ell }_1$-seminorm topologies
• Closure of the cone of sums of 2d-powers in real topological algebras
• Moment problem for symmetric algebras of locally convex spaces
• An intrinsic characterization of moment functionals in the compact case
• On the determinacy of the moment problem for symmetric algebras of a locally convex space
• Projective limits techniques for the infinite dimensional moment problem