Application of localization to the multivariate moment problem II

TL;DR

This paper introduces a new, stronger criterion for determining when a positive semidefinite linear functional on multivariate polynomials corresponds to a positive measure, extending previous results and elucidating the measure's support.

Contribution

It presents a novel criterion that improves upon Nussbaum's, similar to Schmudgen's, and enhances understanding of measure support via localization techniques.

Findings

New criterion is stronger than Nussbaum's

Criterion relates measure support to quadratic modules

Extends Lasserre's results on measure support

Abstract

The paper is a sequel to the paper "Application of localization to the multivariate moment problem" by the same author. A new criterion is presented for a positive semidefinite linear functional on the real polynomial algebra to correspond to a positive Borel measure on real n-space. The criterion is stronger than Nussbaum's criterion and is similar in nature to a criterion of Schmudgen. It is also explained how the criterion allows one to understand the support of the associated measure in terms of the non-negativity of the linear functional on a quadratic module of the real polynomial algebra. This latter result extends a result of Lasserre. The techniques employed are the same localization techniques employed already in two earlier papers by the same author.