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Tomasz Beberok

Publications and source records attributed to Tomasz Beberok.

14 recordsLinked to original sources
arXiv2025

On Bernstein inequalities on the unit ball

Two types of Bernstein inequalities are established on the unit ball in $\mathbb{R}^d$, which are stronger than those known in the literature. The first type consists of inequalities in $L^p$ norm for a fully symmetric doubling weight on the unit ball. The second type consists of sharp inequalities in $L^2$ norm for the Jacobi weight, which are established via a new self-adjoint form of the spectral operator that has orthogonal polynomials as eigenfunctions.

math.CA↗
arXiv2024

On the Lebesgue constant of the Morrow-Patterson points

The study of interpolation nodes and their associated Lebesgue constants are central to numerical analysis, impacting the stability and accuracy of polynomial approximations. In this paper, we will explore the Morrow-Patterson points, a set of interpolation nodes introduced to construct cubature formulas of a minimum number of points in the square for a fixed degree $n$. We prove that their Lebesgue constant growth is ${\cal O}(n^2)$ as was conjectured based on numerical evidence about twenty years ago in the paper by Caliari, M., De Marchi, S., Vianello, M., {\it Bivariate polynomial interpolation on the square at new nodal sets}, Appl. Math. Comput. 165(2) (2005), 261--274.

math.NA↗
arXiv2016

Hilbert-Schmidt Hankel operators over semi-Reinhardt domains

Let $Ω$ be an arbitrary bounded semi-Reinhardt domain in $\mathbb{C}^{m+n}$. We show that for $m \geq 2$, if a Hankel operator with an anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $L_a^2(Ω)$, then it must equal zero. This fact has previously been proved for Reinhardt domains.

math.CV↗
arXiv2015

The Bergman kernel for intersection of two complex ellipsoids

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for intersection of two complex ellipsoids $\{z \in \mathbb{C}^3 \colon |z_1|^p + |z_2|^q < 1, \quad |z_1|^p + |z_3|^r < 1\}$. We consider cases $p=6, q= r= 2$ and $p=q=r=2$. We also investigate the Lu Qi-Keng problem for $p=q=r=2$.

math.CV↗
arXiv2015

Explicit formulas of the Bergman kernel for some Reinhardt domains

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains $\{|z_3|^λ < |z_1|^{2p} + |z_2|^2, \quad |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and $\{|z_4|^λ < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.

math.CV↗