One generalization of the classical moment problem (original) (raw)

Let astP\ast_PastP be a product on lrmfinl_{\rm{fin}}lrmfin (a space of all finite sequences) associated with a fixed family (Pn)n=0infty(P_n)_{n=0}^{\infty}(Pn)n=0infty of real polynomials on mathbbR\mathbb{R}mathbbR. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of astP\ast_PastP-positive functionals on lrmfinl_{\rm{fin}}lrmfin. If (Pn)n=0infty(P_n)_{n=0}^{\infty}(Pn)n=0infty is a family of the Newton polynomials Pn(x)=prodi=0n−1(x−i)P_n(x)=\prod_{i=0}^{n-1}(x-i)Pn(x)=prodi=0n−1(x−i) then the corresponding product star=astP\star=\ast_Pstar=astP is an analog of the so-called Kondratiev--Kuna convolution on a ``Fock space''. We get an explicit expression for the product star\starstar and establish a connection between star\starstar-positive functionals on lrmfinl_{\rm{fin}}lrmfin and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).