We show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems.

Keywords: Bernstein operators Pointwise approximation Rate of convergence a b s t r a c t In the present paper, we study pointwise approximation by Bernstein–Durrmeyer type operators in the mobile interval x 2 0; 1 À 1 nþ1 h i , with use of Peetre's K-functional and x 2 u k ðf ; tÞ ð0 6 k 6 1Þ, we give its properties and obtain the direct and inverse theorems for these operators.

In this paper, when approximating a continuos non-negative function on the unit interval, we present an alternative way to the classical Bernstein polynomials. Our new operators become nonlinear, however, for some classes of functions, they provide better error estimations than the Bernstein polynomials. Furthermore, we obtain a simultaneous approximation result for these operators. 2010 Mathem...

We constructed a summation–integral type operator based on the latest research in linear positive operators area. establish some approximation properties for this new operator. highlight qualitative part of presented operator; we studied uniform convergence, Voronovskaja-type theorem, and Grüss–Voronovskaja result. Our subsequent study focuses direct theorem using Ditzian–Totik modulus smoothne...

Continuous set-valued functions with convex images can be approximated by known positive operators of approximation, such as the Bernstein polynomial operators and the Schoenberg spline operators, with the usual sum between numbers replaced by the Minkowski sum of sets. Yet these operators fail to approximate set-valued functions with general sets as images. The Bernstein operators with growing...

Keywords: q-Bernstein–Schurer–Stancu operators q-integers Modulus of smoothness Rate of convergence a b s t r a c t In the present paper, we introduce the Stancu type generalization of the Bernstein–Schurer operators based on q integers. First, we prove the basic convergence of S ða;bÞ n;p ð:; q; xÞ and then obtain the rate of convergence by these operators in terms of the modulus of continuity...