= de la Vallée Poussin sum
{c}
{title2=$v_{n,m}f=\frac1m\sum_{j=n}^{n+m-1}s_jf$}
= de la Vallée Poussin mean
{c}
{synonym}
This average of <Fourier partial sums> reproduces every degree-at-most-$n$ <trigonometric polynomial>. With $\sigma_r=r^{-1}\sum_{j=0}^{r-1}s_j$,
$$
v_{n,m}=\frac{(n+m)\sigma_{n+m}-n\sigma_n}{m},\qquad
\|v_{n,m}\|_\infty\le1+\frac{2n}{m}.
$$
The bound follows because <Fejér summation is a uniform-norm contraction>. For $n=0$, the term involving $\sigma_0$ is omitted.
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