arXiv subjects

Roman Bessonov

Publications and source records attributed to Roman Bessonov.

9 recordsLinked to original sources

arXiv2026

From local to global asymptotic behaviour of orthogonal polynomials

Let $\{\phi^*_n\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $\mu$ of Szeg\H{o} class, and let $D_{\mu}$ be the Szeg\H{o} function of $\mu$. We prove the uniform Ces\`aro asymptotics $$ \sup_{z \in \Gamma_\zeta}\Biggl(\frac{1}{n}\sum_{k = 0}^{n-1}\Bigl||\phi_k^*(z) D_{\mu}(z)|^2 - 1\Bigr|\Biggr) \to 0, \qquad n \to \infty, $$ for almost all Stolz angles $\Gamma_{\zeta}$, $\zeta\in \partial \mathbb{D}$. This extends a well-known asymptotic result of M\'at\'e, Nevai, and Totik (1991) from the local scale $O(1/n)$ near $\partial \mathbb{D}$ to the global scale $O(1)$. We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szeg\H{o} using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels under various assumptions on the orthogonality measure.

math.CA↗

arXiv2025

Direct and inverse spectral continuity for Dirac operators

The half-line Dirac operators with $L^2$-potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general $L^2$-case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with $\delta$-interactions on a half-lattice in terms of the Schur's algorithm for analytic functions.

math.SP↗

arXiv2023

On rate of convergence for universality limits

Given a probability measure $\mu$ on the unit circle $\mathbb{T}$, consider the reproducing kernel $k_{\mu,n}(z_1, z_2)$ in the space of polynomials of degree at most $n-1$ with the $L^2(\mu)$-inner product. Let $u, v \in \mathbb{C}$. It is known that under mild assumptions on $\mu$ near $\zeta \in \mathbb{T}$, the ratio $k_{\mu,n}(\zeta e^{u/n}, \zeta e^{v/n})/k_{\mu,n}(\zeta, \zeta)$ converges to a universal limit $S(u, v)$ as $n \to \infty$. We give an estimate for the rate of this convergence for measures $\mu$ with finite logarithmic integral.

math.CV↗

arXiv2020

Reflectionless canonical systems, II. Almost periodicity and character-automorphic Fourier transforms

We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions on a compact Riemann surface, the construction is based on reproducing kernels of character-automorphic Hardy spaces in Widom domains with respect to Martin measure. We also construct unitary character-automorphic Fourier transforms which generalize the Paley-Wiener theorem. Finally, we find the correct notion of almost periodicity which holds for canonical system parameters in Arov gauge, and we prove generically optimal results for almost periodicity for Potapov-de Branges gauge, and Dirac operators.

math.SP↗

arXiv2020

Reflectionless canonical systems, I. Arov gauge and right limits

In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur function along the flow of associated boundary values at infinity. In addition to results in Arov gauge, this provides a gauge-independent perspective on the Krein-de Branges formula and the reflectionless property of right limits on the absolutely continuous spectrum. This work has applications to inverse spectral problems which have better behavior with respect to a normalization at an internal point of the resolvent domain.

math.SP↗

arXiv2019

Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Let $\mu$ be a measure from Szeg\H{o} class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $\mu$. In this paper, we prove a version of the classical Szeg\H{o}'s formula which controls the oscillation of $f_n$ on $\mathbb T$ for all $n \ge 0$. Then, we focus on an analog of Lusin's conjecture for polynomials $\{\varphi_n\}$ orthogonal with respect to measure $\mu$ and prove that pointwise convergence of $\{|\varphi_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $\varphi_n$.

math.CV↗

arXiv2019

De Branges canonical systems with finite logarithmic integral

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szego theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein-Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

math.CA↗

arXiv2015

Averaged wave operators and complex-symmetric operators

We study the behaviour of sequences $U_2^n X U_1^{-n}$, where $U_1, U_2$ are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator $XU_1-U_2X$ is small in a sense. The conjecture about the weak averaged convergence of the difference $U_2^n X U_1^{-n}-U_2^{-n} X U_1^n$ to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where $U_1=U_2$ is the unitary operator of multiplication by $z$ on $L^2(\mu)$, sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.

math.FA↗

arXiv2010

Symbols of truncated Toeplitz operators

We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: 1) properties of truncated Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by Donald Sarason.

math.FA↗