One generalization of the classical moment problem

Volodymyr Tesko

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

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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).

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