Genuine q-Stancu-Bernstein-Durrmeyer Operators

Abstract

In the present paper, we introduce the genuine q-Stancu-Bernstein-Durrmeyer operators Z(n)(q,alpha)(f;x). We calculate the moments of these operators, Z(n)(q,alpha)(tj;x) for j=0,1,2, which follows a symmetric pattern. We also calculate the second order central moment Z(n)(q,alpha)((t-x)2;x). We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein-Durrmeyer operators in terms of Lipschitz-type maximal function of order beta and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order.