Some approximation properties of a certain nonlinear Bernstein operators

Abstract

The present paper concerns with the introduction a certain sequence of nonlinear Bernstein operators NB_{n}f of the form

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

acting on bounded functions on an interval [0,1], where P_{k,n} satisfy some suitable assumptions. We will also investigate the pointwise convergence of this operators in some functional spaces. As a result, this study can be considered as an extension of the results dealing with the linear Bernstein Polynomials. As far as we know this kind of study is the first one on the nonlinear Bernstein approximation operators.