Asymptotic properties of powers of linear positive operators which preserve e2

Abstract

In this paper we prove Korovkin type theorem for iterates of general positive linear operators T : C[0, 1] -> C[0, 1] which preserve e(2) and derive quantitative estimates in terms of moduli of smoothness. The results can be applied to several well-known operators; we present here the Bernstein, the q-Bernstein, the genuine Bernstein-Durrmeyer and the genuine q-Bernstein-Durrmeyer operators. (C) 2011 Elsevier Ltd. All rights reserved.