Direct and converse results for multivariate generalized Bernstein polynomials

Abstract

In this paper, the following generalization of multivariate Bernstein polynomials has been studied: B n ( f ; x 1 , x 2 , … , x m ) = ∑ k 1 , k 2 , … , k m = 0 n f k 1 b n , k 2 b n , … , k m b n ∏ j = 1 m n k j x j k j ( 1 - x j ) n - k j , where ( b n ) is a sequence of positive numbers such that lim n → ∞ n / b n = 1 and direct and inverse theorems for this polynomials have been presented.