Moment problem in infinitely many variables

TL;DR

This paper extends the multivariate moment problem to infinitely many variables, providing sharp results, extending classical theorems, and generalizing support descriptions in the context of polynomial algebras.

Contribution

It generalizes the moment problem to infinitely many variables, extending classical theorems and support descriptions, using an advanced localization method.

Findings

Extended Haviland's and Nussbaum's theorems to infinite variables.

Validated Lasserre's support description in this broader setting.

Provided sharp conditions for the moment problem in countably infinite variables.

Abstract

The multivariate moment problem is investigated in the general context of the polynomial algebra $\mathbb{R}[x_i \mid i \in \Omega]$ in an arbitrary number of variables $x_i$, $i\in \Omega$. The results obtained are sharpest when the index set $\Omega$ is countable. Extensions of Haviland's theorem [Amer. J. Math., 58 (1936) 164-168] and Nussbaum's theorem [Ark. Math., 6 (1965) 179-191] are proved. Lasserre's description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of $\mathbb{R}[x_i \mid i \in \Omega]$ in [Trans. Amer. Math. Soc., 365 (2013) 2489-2504] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author.