Moment problems in an infinite number of variables (Q2790329)

The authors investigate the \(K\)-moment problem in an infinite number of variables. Specifically, setting \(\mathbb{R}^\infty= \bigtimes_{d=1}^\infty\mathbb{R}\), giving a closed set \(K\subset\mathbb{R}^\infty\) and a map \(s:S_*\mapsto\mathbb{R}\), where \(S_*\) denotes the set of infinite tuples consisting of nonnegative integers with nonnull entries only for a finite number of indices, necessary and sufficient conditions are looked for, to insure the existence and the uniqueness of a (positive) measure \(\sigma\) on \(\mathbb{R}^\infty\), with support in \(K\), such that \(s(n)=\int_{\mathbb{R}^\infty} x^nd\sigma(x)\), \(n\in S_*\). The existence of such a \(\sigma\) is insured by an exension of the classical Haviland theorem, yielding the positivity of the corresponding Riesz functional, via the positivity of the restrictions of polynomials to the set \(K\). Using an analogue of \textit{K. Schmüdgen}'s theorem [Math. Nachr. 88, 385--390 (1979; Zbl 0424.46041)], the authors also obtain a sufficient condition for the uniqueness of the measure \(\sigma\). An application for stochastic processes is obtained, too.