Moment problem in infinitely many variables

الأصل · EN

The multivariate moment problem is investigated in the general context of the polynomial algebra R[xᵢ i ∈ Ω] in an arbitrary number of variables xᵢ, i∈ Ω. The results obtained are sharpest when the index set Ω 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 R[xᵢ i ∈ Ω] 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.