Positive definite kernels satisfying difference equations - Zagorodnyuk, S. M. (Methods of Functional Analysis and Topology, 2010)
We study positive definite kernels K = (K_n,m)_n,mA, A=Z or A=Z_+, which satisfy a difference equation of the form L_n K = L_m K, or of the form L_n 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.