Open Access

# Positive definite kernels satisfying difference equations

### Abstract

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.

### Article Information

 Title Positive definite kernels satisfying difference equations Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 83-100 MathSciNet MR2656134 Copyright The Author(s) 2010 (CC BY-SA)

### Authors Information

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

### Citation Example

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

### BibTex

@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 = {http://mfat.imath.kiev.ua/article/?id=520},
}