On the rates of pointwise convergence for Bernstein polynomials

Abstract

Let f be a real bounded function defined on the interval [0, 1], which is affine on a subinterval (a,b) of [0,1], and let Bnf be its associated nth Bernstein polynomial. We prove that, for any x in (a,b), |Bnf(x)-f(x)| converges to 0 as n tends to infinity at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of (a, b). For Bernstein–Kantorovich type operators similar properties hold, whenever f is assumed to be constant instead of affine.