# Zagorodnyuk S. M.

### Orthogonal polynomials related to some Jacobi-type pencils

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1180-1190

We study a generalization of the class of orthonormal polynomials on the real axis. These polynomials satisfy the following relation: $(J_5 \lambda J_3)\vec{p}(\lambda) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec{p}(\lambda) = (p_0(\lambda ), p_1(\lambda ), p_2(\lambda ),...)^T$, the superscript $T$ denotes the operation of transposition with the initial conditions $p_0(\lambda ) = 1,\; p_1(\lambda) = \alpha \lambda + \beta,\; \alpha > 0, \beta \in R$. Certain orthonormality conditions for the polynomials $\{ pn(\lambda )\}^{\infty}_n = 0$ are obtained. An explicit example of these polynomials is constructed.

### Nevanlinna formula for the truncated matrix trigonometric moment problem

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1053-1066

This paper is a continuation of our investigation on the truncated matrix trigonometric moment problem begun in Ukr. Mat. Zh. - 2011. - 63, № 6. - P. 786-797. In the present paper, we obtain the Nevanlinna formula for this moment problem in the general case. We assume here that there is more than one moment and the moment problem is solvable and has more than one solution. The coefficients of the corresponding matrix linear fractional transformation are expressed in explicit form via prescribed moments. Simple determinacy conditions for the moment problem are presented.

### Truncated matrix trigonometric problem of moments: operator approach

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 786-797

We study the truncated matrix trigonometric moment problem. We obtain parametrization of all solutions of this moment problem (in both nondegenerate and degenerate cases) via an operator approach. This parametri-zation establishes a one-to-one correspondence between a certain class of analytic functions and all solutions of the problem. We use important results on generalized resolvents of isometric operators, obtained by M. E. Chumakin.

### On the strong matrix Hamburger moment problem

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 471–482

We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators.