One generalization of the classical moment problem

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 a connection between $\star$-positive functionals on $l_{\mathrm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).

Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=1

References in corpus (1)

• Non-equilibrium stochastic dynamics of continuous systems and Bogoliubov generating functionals