The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric

Abstract

We investigate the approximation of functions by Bernstein polynomials. We prove that

$$^\tau [0,1]^{(f,B_n (f))} \leqslant \mu _f \left( {4\sqrt {\tfrac{{\ln n}}{n}} } \right) + \left( {4\sqrt {\tfrac{{\ln n}}{n}} } \right),$$

((1))

where r_[0,1](f, Bn(f)) is the Hausdorff distance between the functionsf(x) and B_n(f; x) in [0,1],

is the modulus of nonmonotonicity off(x). The bound (1) is of better order than that obtained by Sendov. We show that the order of (1) cannot be improved.