Solution
= 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 $j\geq n$. Averaging the sums with $j=n,\ldots,n+m-1$ gives
$$
\boxed{v_{n,m}(t_n)=\frac1m\sum_{j=n}^{n+m-1}t_n=t_n.}
$$
Thus the <de la Vallée Poussin sum> reproduces the entire degree-at-most-$n$ approximation space, including when the actual degree is smaller.