Bernstein-Durrmeyer operators

In this paper, we consider the Durrmeyer-type modifications of the classical Bernstein, Szász and Baskakov operators, and deal with two different kind of problems. Firstly, we obtain new results concerning preservation of shape properties, Lipschitz constants and global smoothness, as well as monotonic convergence under convexity. To do this, we use a probabilistic approach based on representations of these operators in terms of stochastic processes having a.s. nondecreasing paths and satisfying a suitable martingale-type condition. Secondly, we show that the Szász-Durrmeyer operator is the limit, in an appropriate sense, of both the Bernstein-Durrmeyer and the Baskakov-Durrmeyer operators. We provide rates of convergence which are derived from the bounds for the total variation distance between the probability measures involved.