Direct estimate for Bernstein polynomials (Q1335050)

The following pointwise approximation for the Bernstein polynomials \(B_ n (f,x)= \sum_{k=0}^ n {\binom nk} x^ k (1-x)^{n -k} f(k/n)\) are proved: \[ | B_ n (f,x)- f(x)|\leq C\omega^ 2_{\varphi^ \lambda} (f, n^{-1/2} \varphi (x)^{1- \lambda}), \qquad 0\leq \lambda\leq 1, \quad \varphi(x)^ 2= x(1-x). \] This estimates yields a unified treatment of the classical estimate \((\lambda=0)\) and the norm estimate \((\lambda=1)\).