One generalization of the classical moment problem (Q2896657)

Let \(*_P\) be a product on the space \(l_{\text{fin}}\) of all finite sequences of complex numbers associated with a fixed family \(\{ P_n\}_{n=0}^\infty\) of real polynomials on \(\mathbb R\). The paper is devoted to moment-type properties of \(*_P\)-positive functionals on \(l_{\text{fin}}\). The author uses Berezanskii's method of generalized eigenfunction expansions [\textit{J. M. Berezanskiĭ}, Expansions in eigenfunctions of selfadjoint operators. Providence: AMS (1968; Zbl 0157.16601)]. If \(\{P_n\}_{n=0}^\infty\) is a family of the Newton polynomials \(P_n(x)=x(x-1)\cdots (x-n+1)\), then the corresponding product \(*_P\) is similar to the Kondratiev-Kuna convolution [\textit{Y. G. Kondratiev} and \textit{T. Kuna}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, No. 2, 201--233 (2002; Zbl 1134.82308)]. The author gets an explicit expression for this product and finds a connection between \(*_P\)-positive functionals on \(l_{\text{fin}}\) and one-dimensional analogs of the N. N. Bogoliubov's generating functionals used to define correlation functions for systems of statistical mechanics [\textit{N. N. Bogolyubov}, Studies in statistical mechanics. Vol. 1, 1--118 (1962; Zbl 0116.45101)].