Upper estimates in direct inequalities for Bernstein-type operators

We obtain explicit upper estimates in direct inequalities with respect to t he usual sup-norm distance for Bernstein-type operators. Our approach combi nes analytical and probabilistic techniques based on representations of the operators in terms of stochastic processes. We illustrate our results by c onsidering some classical families of operators, such as Weierstrass. Szasz , and Bernstein operators. (C) 2001 Academic Press.