A note on derivatives of Bernstein polynomials (Q1332294)

Take \(f\in C[0,1]\) with the smoothness \(\omega_ r(f,t)= O(t^ \beta)\) for some \(\beta> 0\) and some \(r\in \mathbb{N}\). The author proves that in this case the \(r\)th derivative of the Bernstein polynomial \(B_ n f\) satisfies the estimate \(|(B_ n f)^{(r)}(x)|\leq M_ f\left[{n\over x(1- x)}\right]^{{r-\alpha\over 2}}\) for \(0< \alpha< r\) if and only if \(\omega_ r(f,t)= O(t^ \alpha)\). This extends two earlier results by Ditzian and the author respectively. The proof uses linear combinations of Bernstein polynomials. For Kantorovich polynomials on \(C[0,1]\) the additional smoothness of \(f\) is not necessary.