Solution
= Solution
Let $m$ be a positive integer and let the <trigonometric polynomial> $t_n$ have frequencies only in $[-n,n]$. For every $j\ge n$, its <Fourier partial sum> is the <polynomial> itself: $s_j(t_n)=t_n$. Every term in the defining average of the <de la Vallée Poussin sum> therefore equals $t_n$, giving
$$
\boxed{v_{n,m}(t_n)=t_n}.
$$
This is exact reproduction of the degree-at-most-$n$ <trigonometric polynomials>, irrespective of the positive averaging length $m$.