One generalization of the classical moment problem

Volodymyr Tesko

Methods Funct. Anal. Topology 17 (2011), no. 4, 356-380

Abstract

Let $\ast_P$ be a product on $l_{\mathrm{fin}}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{\infty}$ of real polynomials on $\mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $\ast_P$-positive functionals on $l_{\mathrm{fin}}$.

If $(P_n)_{n=0}^{\infty}$ is a family of the Newton polynomials $P_n(x)=\prod_{i=0}^{n-1}(x-i)$ then the corresponding product $\star=\ast_P$ is an analog of the so-called Kondratiev—Kuna convolution on a “Fock space”. We get an explicit expression for the product $\star$ and establish the connection between $\star$-positive functionals on $l_{\mathrm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals define correlation functions for statistical mechanics systems).

Full Text