A generalized Bernstein approximation theorem

The present paper is concerned with some generalizations of Bernstein’s approximation theorem. One of the most elegant and elementary proofs of the classic result, for a function f(x) defined on the closed interval [0, 1], uses the Bernstein’s polynomials of f,

We shall concern the m-dimensional generalization of the Bernstein’s polynomials and the Bernstein’s approximation theorem by taking an (m−1)-dimensional simplex in cube [0, 1]m. This is motivated by the fact that in the field of mathematical biology naturally arouse dynamic systems determined by quadratic mappings of “standard” (m − 1)-dimensional simplex to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the m-dimensional generalizations of the Bern- stein’s polynomials and the Bernstein’s approximation theorem.