One generalization of the classical moment problem

TL;DR

This paper generalizes the classical moment problem by exploring moment-type properties of positive functionals on finite sequences with a new product structure linked to polynomial families, including Newton polynomials, and connects these to convolution and generating functionals.

Contribution

It introduces a new product on finite sequences associated with polynomial families, extending the moment problem framework and linking it to convolution structures and generating functionals.

Findings

Explicit expression for the product 

Connection between positive functionals and Bogoliubov functionals

Extension of the moment problem to polynomial-based products

Abstract

Let $\ast_P$ be a product on $l_{\rm{fin}}$ (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 $l_{\rm{fin}}$. 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 $l_{\rm{fin}}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).