Application of localization to the multivariate moment problem II

Abstract

The paper is a sequel to the paper [5], Math. Scand. 115 (2014), 269--286, by the same author. A new criterion is presented for a PSD linear map $L \colon \mathbb{R}[\underline{x}] \to \mathbb{R}$ to correspond to a positive Borel measure on $\mathbb{R}^n$. The criterion is stronger than Nussbaum's criterion (Ark. Math. 6 (1965), 171--191) and is similar in nature to Schmüdgen's criterion in Marshall [5] and Schmüdgen, Ark. Math. 29 (1991), 277--284. It is also explained how the criterion allows one to understand the support of the associated measure in terms of the non-negativity of $L$ on a quadratic module of $\mathbb{R}[\underline{x}]$. This latter result extends a result of Lasserre, Trans. Amer. Math. Soc. 365 (2013), 2489--2504. The techniques employed are the same localization techniques employed already in Marshall (Cand. Math. Bull. 46 (2003), 400--418, and [5]), specifically one works in the localization of $\mathbb{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$.