Approximation properties of complex q-Szasz-Mirakjan operators in compact disks

Abstract

This paper deals with approximating properties of the newly defined q-generalization of the Szasz-Mirakjan operators in the case q > 1. Quantitative estimates of the convergence, the Voronovskaja's theorem and saturation of convergence for complex q-Szasz-Mirakjan operators attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in {z is an element of C : vertical bar Z vertical bar < R}, R > 2q, the rate of approximation by the q-Szasz-Mirakjan operators (q > 1) is of order q(-n) versus 1/n for the classical Szasz-Mirakjan operators. (C) 2010 Elsevier Ltd. All rights reserved.