On convergence of certain nonlinear Bernstein operators

Abstract

In this article, we concern with the nonlinear Bernstein operators NBnf of the form (NBnf)(x)= n?k=0 Pn,k (x,f (k/n)), 0 ? x ? 1, n?N, acting on bounded functions on an interval [0,1], where Pn,k satisfy some suitable assumptions. As a continuation of the very recent paper of the authors [22], we estimate their pointwise convergence to a function f having derivatives of bounded (Jordan) variation on the interval [0,1]. We note that our results are strict extensions of the classical ones, namely, the results dealing with the linear Bernstein polynomials.