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

A trigonometric polynomial $t_n$ of degree at most $n$ has zero Fourier coefficients outside $[-n,n]$. Its Fourier partial sum $s_j(t_n)$ therefore equals $t_n$ for every $j\ge n$. Averaging the sums with $j=n,\dots ,n+m-1$ givesThus the de la Vallée Poussin sum reproduces the entire degree-at-most-$n$ approximation space, including when the actual degree is smaller.