Optimal approximation class for multivariate Bernstein operators

Abstract

For the Bernstein polynomial approximation process on a simplex or a cube, the class of functions yielding optimal approximation will be given. That is, we will find the class of functions for which ∥B n f − f∥ C(S) = O(n −1 ) in terms of the behaviour of a certain K-functional. Moreover, this is done in the context of direct and converse results which fields an improvement on such results as well