The Moment Problem for Continuous Positive Semidefinite Linear Functionals

Abstract

Let τ be a locally convex topology on the countable dimensional polynomial ${\mathbb{R}}$ -algebra ${\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}$ . Let K be a closed subset of ${\mathbb{R} ^{n}}$ , and let ${M := M_{{g_1, \ldots, g_s}}}$ be a finitely generated quadratic module in ${\mathbb{R} [\underline{X}]}$ . We investigate the following question: When is the cone Psd(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 ${M = \sum \mathbb{R} [\underline{X}]^{2}}$ with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron.