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Positive definite kernels satisfying difference equations


We study positive definite kernels $K = (K_{n,m})_{n,m\in A}$, $A=\mathbb Z$ or $A=\mathbb Z_+$, which satisfy a difference equation of the form $L_n K = \overline L_m K$, or of the form $L_n \overline L_m K = K$, where $L$ is a linear difference operator (here the subscript $n$ ($m$) means that $L$ acts on columns (respectively rows) of $K$). In the first case, we give new proofs of Yu.M. Berezansky results about integral representations for $K$. In the second case, we obtain integral representations for $K$. The latter result is applied to strengthen one our result on abstract stochastic sequences. As an example, we consider the Hamburger moment problem and the corresponding positive matrix of moments. Classical results on the Hamburger moment problem are derived using an operator approach, without use of Jacobi matrices or orthogonal polynomials.

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TitlePositive definite kernels satisfying difference equations
SourceMethods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 83-100
MathSciNet   MR2656134
CopyrightThe Author(s) 2010 (CC BY-SA)

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S. M. Zagorodnyuk
School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody sq., Kharkiv, 61077, Ukraine

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S. M. Zagorodnyuk, Positive definite kernels satisfying difference equations, Methods Funct. Anal. Topology 16 (2010), no. 1, 83-100.


@article {MFAT520,
    AUTHOR = {Zagorodnyuk, S. M.},
     TITLE = {Positive definite kernels satisfying difference equations},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {16},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {83-100},
      ISSN = {1029-3531},
  MRNUMBER = {MR2656134},
       URL = {},

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