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One generalization of the classical moment problem

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Abstract

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

Key words: Convolution, positive functional, moment problem, projection spectral theorem, Sheffer polynomials.


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Article Information

TitleOne generalization of the classical moment problem
SourceMethods Funct. Anal. Topology, Vol. 17 (2011), no. 4, 356-380
MathSciNet MR2907364
MilestonesReceived 11/07/2011
CopyrightThe Author(s) 2011 (CC BY-SA)

Authors Information

Volodymyr Tesko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine


Citation Example

Volodymyr Tesko, One generalization of the classical moment problem, Methods Funct. Anal. Topology 17 (2011), no. 4, 356-380.


BibTex

@article {MFAT1,
    AUTHOR = {Tesko, Volodymyr},
     TITLE = {One generalization of the classical moment problem},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {17},
      YEAR = {2011},
    NUMBER = {4},
     PAGES = {356-380},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1},
}


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