Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 70 / 2 / a / Solution

The Bernstein polynomial is the positive weighted averageFor $0\le x\le 1$, the Bernstein basis weights are nonnegative and the binomial theorem gives ${\sum }_{k=0}^n{b}_{n,k}(x)=1$. Hence $|B_n(f,x)|\le {\sum }_k\Vert f{\Vert }_{\infty }{b}_{n,k}(x)=\Vert f{\Vert }_{\infty }$. In particular,This is the contraction property of the positive linear operator on continuous functions defined by the Bernstein polynomial.