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Rostyslav Kozhan

Publications and source records attributed to Rostyslav Kozhan.

At least 19 recordsLinked to original sources
arXiv2026

Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures

Given multiple orthogonal polynomials on the real line with respect to a system $\bmμ = (μ_1,\ldots,μ_r)$, we investigate multiple orthogonal polynomials associated with any rational perturbation of the form $$ \widetilde{\bmμ}=\Big(\frac{Φ_1}{Ψ_{1}} μ_1,\dots,\frac{Φ_r}{Ψ_r}μ_r\Big), $$ for any polynomials $Φ_1,\dots,Φ_r$ and $Ψ_1,\dots,Ψ_r$. We derive the analogues of Uvarov's determinantal formula for the multiple orthogonal polynomials of type I and type II for $\widetilde{\bmμ}$ and establish necessary and sufficient condition for normality of the indices. The result allows the polynomials $\{Φ_j,Ψ_j\}_{j=1}^r$ to be arbitrary and permits the addition of finitely many point masses to each of the measures $μ_j$. Moreover, the measures $μ_j$ may be taken as quasi-definite linear functionals, which is of interest even in the case $r=1$.

math.CA↗
arXiv2026

Zeros of Laurent multiple orthogonal polynomials on the unit circle

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within the complex open unit disk for any Angelesco or AT system. For the second formulation, we establish normality of all indices of the form $(\bm{n};\bm{n})$, $(\bm{n}+\bm{e}_j;\bm{n})$, and $(\bm{n};\bm{n}+\bm{e}_j)$ for any Angelesco or AT system, thereby enabling the full application of the Szegő mapping and Geronimus relations from arXiv:2601.04783 in the multiple orthogonality setting.

math.CV↗
arXiv2026

Szegő Mapping and Hermite--Padé Polynomials for Multiple Orthogonality on the Unit Circle

We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these polynomials can be characterized as solutions of a general two-point Hermite--Padé approximation problem. We derive Szegő-type recurrence relations, establish compatibility conditions for the associated recurrence coefficients, and obtain Christoffel--Darboux formulas as well as Heine-type determinantal representations. Furthermore, by extending the Szegő mapping and the Geronimus relations, we relate these Laurent multiple orthogonal polynomials to multiple orthogonal polynomials on the real line, thereby making explicit the connection between multiple orthogonality on the unit circle and on the real line.

math.CA↗
arXiv2026

Angelesco and AT systems on the Unit Circle

We introduce the concept of Laurent multiple orthogonality on the unit circle and define Angelesco and AT systems in this setting. Using a generalized Andreief identity, we establish normality of all multi-indices for any such system, thereby ensuring existence and uniqueness of Laurent multiple orthogonal polynomials of type I and type II at every location. As an application, we demonstrate existence and uniqueness of the approximants for two natural two-point Hermite-Padé problems -- type I and type II -- arising in the simultaneous rational approximation of $r$ Carathéodory functions.

math.CA↗
arXiv2025

Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $α_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients $α_{n,N}$. Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients.

math.CA↗
arXiv2025

Christoffel Transform and Multiple Orthogonal Polynomials

We investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. We apply these results to show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomials corresponding to its one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros. We also identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms.

math.CA↗
arXiv2025

Zeros of Multiple Orthogonal Polynomials: Location and Interlacing

We prove a criterion on the possible locations of zeros of type I and type II multiple orthogonal polynomials in terms of normality of degree $1$ Christoffel transforms. We provide another criterion in terms of degree $2$ Christoffel transforms for establishing zero interlacing of the neighbouring multiple orthogonal polynomials of type I and type II. We apply these criteria to establish zero location and interlacing of type I multiple orthogonal polynomials for Nikishin systems. Additionally, we recover the known results on zero location and interlacing for type I multiple orthogonal polynomials for Angelesco systems, as well as for type II multiple orthogonal polynomials for Angelesco and AT systems. Finally, we demonstrate that normality of the higher order Christoffel transforms is naturally related to the zeros of the Wronskians of consecutive orthogonal polynomials.

math.CA↗
arXiv2024

Nikishin systems on the unit circle

We introduce Nikishin system of $r$ probability measures on the unit circle. We show that such systems satisfy the AT property and therefore normality, introduced in~\cite{KVMLOPUC}, for any multi-index $(n_1,\ldots,n_r)\in\mathbb{N}^r$ with same-parity components satisfying $n_1 \ge n_2 \ge\ldots\ge n_r$. In the case of $r=2$, we demonstrate that the same property holds without requiring $n_1 \ge n_2 \ge\ldots\ge n_r$. The analogous simple proof works for Nikishin systems on the real line for indices satisfying $n_j\ge \max\{n_{j+1},\ldots,n_r\}-1$, $j=1,\ldots,r-1$. This is related to the proof by Cousseement and Van Assche for $r=2$.

math.CA↗
arXiv2024

Szegő Recurrence for Multiple Orthogonal Polynomials on the Unit Circle

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel$\unicode{x2013}$Darboux formula. These results stand directly in analogue with the nearest neighbour recurrence relations from the real line counterpart.

math.CA↗
arXiv2023

A generalized Hermite-Biehler theorem

The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing is broken at exactly one location. Using this we solve the direct and inverse spectral problem for rank-one multiplicative perturbations of finite Hermitian matrices. We also treat certain rank two additive perturbations of finite Jacobi matrices.

math.CA↗
arXiv2022

Jost asymptotics for matrix orthogonal polynomials on the real line

We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.

math.AP↗
arXiv2021

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others. Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker--Campbell--Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero. We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

math.PR↗
arXiv2020

Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class

A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equation becomes ordinary. For Angelesco systems, the result is illustrated numerically.

math.CA↗
arXiv2018

On Gaussian random matrices coupled to the discrete Laplacian

We study operators obtained by coupling an $n \times n$ random matrix from one of the Gaussian ensembles to the discrete Laplacian. We find the joint distribution of the eigenvalues and resonances of such operators. This is one of the possible mathematical models for quantum scattering in a complex physical system with one semi-infinite lead attached.

math-ph↗
arXiv2017

Relative Szegő asymptotics for Toeplitz determinants

We study the asymptotic behavior, as $n\to\infty$, of ratios of Toeplitz determinants $D_n(e^h dμ)/D_n(dμ)$ defined by a measure $μ$ on the unit circle and a sufficiently smooth function $h$. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on $h$ and only a few Verblunsky coefficients associated to $μ$. As a result, we establish a relative version of the Strong Szegő Limit Theorem for a wide class of measures $μ$ with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

math-ph↗
arXiv2016

Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $β>0$, not merely $β=1,2,4$.

math.PR↗