Convergence of rational Bernstein operators

TL;DR

This paper investigates the convergence behavior and error bounds of rational Bernstein operators, establishing conditions for convergence and providing a Voronovskaja theorem with explicit moment calculations.

Contribution

It introduces new convergence criteria and error estimates for rational Bernstein operators, including a Voronovskaja theorem based on explicit higher order moments.

Findings

Rational Bernstein operators converge if the maximum node difference tends to zero.

Error estimates are derived in terms of the node differences.

A Voronovskaja theorem with explicit moment computations is established.

Abstract

In this paper we discuss convergence properties and error estimates of rational Bernstein operators introduced by P. Pi\c{t}ul and P. Sablonni\`{e}re. It is shown that the rational Bernstein operators R_n converge to the identity operator if and only if \Delta_n, the maximal difference between two consecutive nodes of R_n, is converging to zero. Error estimates in terms of \Delta_n are provided. Moreover a Voronovskaja theorem is presented which is based on the explicit computation of higher order moments for the rational Bernstein operator.