Moment problem in infinitely many 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.

References in corpus (1)

• The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

Cited by in corpus (5)

• The full moment problem on subsets of probabilities and point configurations
• Moment problem for symmetric algebras of locally convex spaces
• The strong truncated Hamburger moment problem with and without gaps
• The singular bivariate quartic tracial moment problem
• Projective limits techniques for the infinite dimensional moment problem