Inverse theorems for multidimensional Bernstein operators

Let B n f be the m-dimensional Bernstein polynomials on a simplex or on a cube. The class of functions for which \ n f -f\\ = O(n~a) is determined. That is, necessary and sufficient conditions on the smoothness of / in the simplex or the cube and especially near their boundaries are given so that \ n f -f\\ = O(n~a). Interpolation of spaces, and in particular the characterization of the interpolation space, is one of the tools used.