On Convergence of Certain Nonlinear Bernstein Operators

Harun Karsli, Ismail U. Tiryaki, H. Erhan Altin

In this article, we concern with the nonlinear Bernstein operators $NB_nf$ of the form \[ (NB_nf)(x)=um_{k=0}^{n}P_{n,k}\Big(x,f\Big(\frac{k}{n}\Big)\Big),\qquad0eq xeq1,\quad nı\mathbb N, \] 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 [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.