A Generalization of Bernstein–Kantorovič Operators

Abstract

In this paper, we use a probabilistic setting to introduce a double sequence (L〈k〉n) of linear polynomial operators which includes, as particular cases, the classical Bernstein operators, the Kantorovič operators, and the operators recently introduced by Cao. For these operators, we discuss several approximation properties. In particular, we deal with the convergence properties according to the way in which the different parameters vary, and the preservation of global smoothness and classes of functions determined by concave moduli of continuity. A remarkable feature of our approach is that if f is differentiable, the approximation properties of both L〈k〉nf and its derivatives can be discussed simultaneously. Throughout the paper, probabilistic methods play an important role.