Bernstein–Schurer–Kantorovich operators based on q-integers

Abstract

In this paper, we introduce a new Kantorovich type generalization of the q-Bernstein–Schurer operators defined in Muraru (2011). First, we give the basic convergence theorem and then obtain the local direct results for these operators, estimating the rate of convergence by using the modulus of smoothness and the Lipschitz type maximal function, respectively. We also obtain a Voronovskaja type theorem and investigate the statistical approximation properties of these operators with the help of a Korovkin type statistical approximation theorem given in Duman (2008).