On convergence of certain nonlinear Bernstein operators

Abstract

The present paper concerns with the very recently introduced nonlinear Bernstein operators NB_{n}f of the form

(NB_{n}f)(x)=∑_{k=0}ⁿP_{n,k}(x,f((k/n))),0≤x≤1,n∈N,

acting on bounded functions on an interval [0,1], where P_{n,k} satisfy some suitable assumptions. As a continuation of the very recent paper of the authors <cite>kta</cite>, 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.