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, B-n(x) = B-n(f)(x) = Sigma(n)(k=0) f (k/n) ((n)(k))x(k) (1 - x)(n-k) 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 {x(i) >= 0, i = 1, ... , m, Sigma(m)(i=1) x(i) = 1} to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the m-dimensional generalizations of the Bernstein's polynomials and the Bernstein's approximation theorem.