The K-moment problem for continuous linear functionals

TL;DR

This paper addresses the K-moment problem for continuous linear functionals on semi-algebraic sets, introducing a weighted norm approach to characterize nonnegative polynomials and provide explicit Positivstellensatz results.

Contribution

It introduces a weighted $ ext{l}_1$-norm framework to solve the K-moment problem for continuous linear functionals on non-compact semi-algebraic sets, with explicit projections and sparse supports.

Findings

Characterization of the $ ext{l}_w$-closure of preordering and quadratic module as the cone of polynomials nonnegative on K.

Solution of the K-moment problem for $ ext{l}_w$-continuous linear functionals.

Explicit form of a canonical $ ext{l}_w$-projection with sparse support.

Abstract

Given a closed (and non necessarily compact) basic semi-algebraic set $K\subseteq R^n$, we solve the $K$-moment problem for continuous linear functionals. Namely, we introduce a weighted $\ell_1$-norm $\ell_w$ on $R[x]$, and show that the $\ell_w$-closures of the preordering $P$ and quadratic module $Q$ (associated with the generators of $K$) is the cone $psd(K)$ of polynomials nonnegative on $K$. We also prove that $P$ an $Q$ solve the $K$-moment problem for $\ell_w$-continuous linear functionals and completely characterize those $\ell_w$-continuous linear functionals nonnegative on $P$ and $Q$ (hence on $psd(K)$). When $K$ has a nonempty interior we also provide in explicit form a canonical $\ell_w$-projection $g^w_f$ for any polynomial $f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $g^w_f$ is very sparse and does not depend on $K$! This enables us to provide an explicit Positivstellensatz on $K$. At last but not least, we provide a simple characterization of polynomials nonnegative on $K$, which is crucial in proving the above results.