Estimates for Iterates of Positive Linear Operators Preserving Linear Functions

Abstract

We consider iterates of certain (general) positive linear operators preserving linear functions and derive quantitative upper estimates in terms of weighted and non-weighted moduli of smoothness and related K-functionals. We show that corresponding lower estimates in terms of the classical moduli are not possible, while for the Ditzian–Totik modulus the situation can be different. The results can be applied to several well-known operators; we present here the Bernstein and the genuine Bernstein–Durrmeyer operators.