The K-moment problem for continuous linear functionals

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 $g_f^w$ for any polynomial $f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $g_f^w$ 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.

Cited by in corpus (3)

• The Moment Problem for Continuous Positive Semidefinite Linear functionals
• Closure of the cone of sums of 2d-powers in real topological algebras
• On the determinacy of the moment problem for symmetric algebras of a locally convex space