Some properties of two-dimensional Bernstein polynomials

AbstractLet Bn,mf be the Bernstein polynomial of two variables, of degree (n, m), corresponding to a function f: S → R, where S is the unit square [0, 1] × [0, 1]. We demonstrate that, if all the partial derivatives of f of the first r orders exist and are continuous in S, then certain polynomials with integral coefficients, Bn, m(q, r−q), ef, converge to (∂r∂xq ∂yr − q)f both in the uniform and Lp-norms. We define Bn, m(q, r−q), ef to be the polynomial (∂r∂xq ∂yr − q)(Bn,mf) with each coefficient replaced by its integral part. An estimate of the order of approximation is also obtained