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S. Kupin

Publications and source records attributed to S. Kupin.

15 recordsLinked to original sources
arXiv2025

On local non-tangential growth of the resolvent of a banded Toeplitz operator

We study the growth of the resolvent of a Hardy--Toeplitz operator $T_b$ with a Laurent polynomial symbol (\emph{i.e., } the matrix $T_b$ is banded), at the neighborhood of a point $w_0\in\partial(\sigma(T_b))$ on the boundary of its spectrum. We show that such growth is inverse linear in some non-tangential domains at the vertex $w_0$, provided that $w_0$ does not belong to a certain finite set on the complex plane.

math.SP↗
arXiv2016

On zeros of analytic functions satisfying non-radial growth conditions

Extending the results of Borichev--Golinskii--Kupin [2009], we obtain refined Blaschke-type necessary conditions on the zero distribution of analytic functions on the unit disk and on the complex plane with a cut along the positive semi-axis satisfying some non-radial growth restrictions.

math.CV↗
arXiv2015

On non-selfadjoint perturbations of infinite band Schrödinger operators and Kato method

Let $ H_0=-\dd+V_0 $ be a multidimensional Schrödinger ope\-rator with a real-valued potential and infinite band spectrum, and $H=H_0+V$ be its non-selfadjoint perturbation defined with the help of Kato approach. We prove Lieb--Thirring type inequalities for the discrete spectrum of $H$ in the case when $V_0\in L^\infty(\br^d)$ and $V\in L^p(\br^d)$, $p>\max(d/2, 1)$.

math.SP↗
arXiv2015

On complex perturbations of infinite band Schrodinger operators

We study a complex perturbation of a self-adjoint infinite band Schrodinger operator (defined in the form sense), and obtain the Lieb--Thirring type inequalities for the rate of convergence of the discrete spectrum of the perturbed operator to the joint essential spectrum of both operators.

math.SP↗
arXiv2011

A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.

math.SP↗
arXiv2010

Multipoint Schur algorithm, II: generalized moment problems, Gaussian processes and prediction

We use nowdays classical theory of generalized moment problems by Krein-Nudelman [1977] to define a special class of stochastic Gaussian processes. The class contains, of course, stationary Gaussian processes. We obtain a spectral representation for the processes from this class and we solve the corresponding prediction problem. The orthogonal rational functions on the unit circle lead to a class of Gaussian processes providing an example for the above construction.

math.SP↗
arXiv2010

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.

math.CA↗
arXiv2005

Inverse scattering problem for a special class of canonical systems and non-linear Fourier integral. Part I: asymptotics of eigenfunctions

An original approach to the inverse scattering for Jacobi matrices was suggested in a recent paper by Volberg-Yuditskii. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that generalizes the case of Sturm-Liouville (Schrödinger) operator.

math-ph↗
arXiv2004

Asymptotics of the orthogonal polynomials for the Szego class with a polynomial weight

Let p(t) be a trigonometric polynomial, non-negative on the unit circle. We say that a measure σbelongs to a polynomial Szego class, if the logarithm of its density is summable over the circle with the weight p(t). For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that these asymptotics holds in L^2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.

math.CA↗
arXiv2004

The Szego class with a polynomial weight

Let p be a trigonometric polynomial, nonnegative on the unit circle $\mathbb{T}$. We say that a measure $σ$ on $\mathbb{T}$ belongs to the polynomial Szego class, if $dσ=sigma'_{ac}dθ+dσ_s$, $σ_s$ is singular, and $p\ln σ'_{ac}$ is summable on $\mathbb{T}$. For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that this asymptotics holds in the $L^2$ sense on the unit circle. As a corollary, we get existense of certain modified wave operators.

math.CA↗