A class of Bernstein-type operators on the unit disk

TL;DR

This paper introduces and analyzes new Bernstein-type operators for functions on the unit disk, including convergence and rate estimates, with numerical comparisons demonstrating their effectiveness.

Contribution

It develops novel Bernstein-type operators on the disk, including transformations and quadrants, with convergence proofs and numerical performance analysis.

Findings

Operators converge for continuous functions

Rate of convergence estimates provided

Numerical examples compare different operators

Abstract

We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyse the bivariate Bernstein-Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein-Stancu and the Bernstein-type operator on disk quadrants.