Convergence of rational Bernstein operators
For approximation theorists, it resolves convergence conditions and provides error bounds for a class of rational operators.
The paper characterizes convergence of rational Bernstein operators, showing they converge to the identity iff the maximal node spacing tends to zero, and provides error estimates and a Voronovskaja theorem.
In this paper we discuss convergence properties and error estimates of rational Bernstein operators introduced by P. Piţul and P. Sablonnière. It is shown that the rational Bernstein operators R_n converge to the identity operator if and only if Δ_n, the maximal difference between two consecutive nodes of R_n, is converging to zero. Error estimates in terms of Δ_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.