Convergence of rational Bernstein operators

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.