Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 67 / 2 / c / Solution

Take any trigonometric polynomial $t\in {\mathcal{T}}_n$. The reproduction property in (a) givesBy (b),Taking the infimum over $t\in {\mathcal{T}}_n$ yields the polynomial reproduction error bound for this operator. With $E_n(f)={\inf }_{t\in {\mathcal{T}}_n}\Vert f-t{\Vert }_{\infty }$ and $n/m\le M$, it becomesThis is a near-best uniform approximation estimate; it follows from reproduction and boundedness without requiring the de la Vallée Poussin sum itself to be a best approximant.