The Moment Problem for Continuous Positive Semidefinite Linear functionals

TL;DR

This paper studies the conditions under which polynomials nonnegative on a set can be approximated by elements of a quadratic module, linking this to measure representation and exploring closures in various topologies.

Contribution

It provides a characterization of the closure of quadratic modules in terms of measure representation and computes these closures for specific topologies and sets.

Findings

Closure of the quadratic module coincides with nonnegative polynomials on certain convex polyhedra.

Provides measure-theoretic interpretation of polynomial positivity and quadratic module approximation.

Computes closures of sums of squares in weighted norm topologies.

Abstract

Let $\tau$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, 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.