Simultaneous approximation by Bernstein polynomials with integer coefficients

Abstract

We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big O-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein polynomials with integer coefficients is saturated. We identify its saturation rate and the trivial class.

Shape preserving properties of the Bernstein polynomials with integer coefficients

“…The operators B n and B n possess the property of simultaneous approximation, that is, the derivatives of B n (f ) and B n (f ) approximate the corresponding derivatives of f in the uniform norm on [0, 1]. This was established in [2,3] under certain necessary and sufficient conditions, as estimates of the rate the convergence were proved. Hence, trivially, under these conditions, if f (r) (x) is strictly positive or negative, then so are ( B n (f )) (r) (x) and ( B n (f )) (r) (x) at least for n large enough, depending on f .…”