Moment problems in an infinite number of variables

Abstract

Let [Formula: see text]. Given a closed set [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of tuples of nonnegative integers [Formula: see text] with [Formula: see text] for finitely many [Formula: see text], the [Formula: see text]-moment problem on [Formula: see text] entails determining whether or not there exists a measure [Formula: see text] on [Formula: see text] so that [Formula: see text] and [Formula: see text] We prove that [Formula: see text] exists if and only if a natural analogue of the Riesz–Haviland functional [Formula: see text] is [Formula: see text]-positive, i.e. if [Formula: see text] is any polynomial which is nonnegative for all [Formula: see text], then [Formula: see text] We will also provide a sufficient condition for [Formula: see text] to be unique, an analogue of a celebrated theorem of K. Schmüdgen and an application to stochastic processes.