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On convergence of certain nonlinear Bernstein operators

Karsli Harun (Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Golkoy Bolu, Turkey)
Tiryaki Ismail U. (Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Golkoy Bolu, Turkey)
Altin Erhan H. (Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, Golkoy Bolu, Turkey)

In this article, we concern with the nonlinear Bernstein operators NBnf of the form (NBnf)(x)= nΣk=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.

Keywords: nonlinear Bernstein operators, bounded variation, (L-ψ) Lipschitz condition, pointwise convergence