One generalization of the classical moment problem - Tesko, V. A. (Methods of Functional Analysis and Topology, 2011)
Let *_P be a product on l_fin (a space of all finite sequences) associated with a fixed family (P_n)_n=0^∞ of real polynomials on R. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of *_P-positive functionals on l_fin. If (P_n)_n=0^∞ is a family of the Newton polynomials P_n(x)=_i=0^n-1(x-i) then the corresponding product =*_P is an analog of the so-called Kondratiev–Kuna convolution on a "Fock space". We get an explicit expression for the product and establish a connection between -positive functionals on l_fin and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).