de la Vallée Poussin sum ($v_{n,m}f=\frac{1}{m}{\sum }_{j=n}^{n+m-1}{s}_{j}f$)

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.