On the Cheney and Sharma OperatorU

In this paper, we consider the double-indexed Bernstein power series operator Pa, t introduced by Cheney and Sharma and related to the classical Laguerre polynomials. We obtain sharp estimates of the first two moments which allow us to prove that, when acting on a continuous function f, the convergence of Pa, tf to f is uniform on the whole interval w0, 1x. Moreover, we show that the rates of convergence depend on the way in which tra goes to 0. We also show that Pa, t preserves monotonicity and global smoothness. Finally, we consider the monotonicity properties of Pa, t with respect to both parameters. To achieve the mentioned results, we use a probabilistic approach based on the representation of the operator in terms of a suitable multi-indexed stochastic process. The path properties and the martingale-type properties of this process are key points to give short proofs.