On discrete q-derivatives of q-Bernstein operators

Abstract

In the present paper, we shall investigate the pointwise approximation properties of the q analog of the Bernstein operators and estimate the rate of pointwise convergence of these operators to the functions f whose q-derivatives are bounded variations on the interval [0, 1]. We give an estimate for the rate of convergence of the operator (B _{n, q} f) at those points x at which the one-sided q- derivatives D_q⁺ f(x), D_q⁻  f(x) exists. We shall also prove that the operator's B _{n, q} f converge to the limit f. As a continuation of the very recent study of the author on the q-Bernstein Durrmeyer operators [10], the present study will be the first study on the approximation of q analogous of the discrete type operators in the space of D_qBV.