Fejér saturation on a Fourier mode
= Fejér saturation on a Fourier mode
{c}
{title2=$\|\sigma_n(\cos kx)-\cos kx\|_\infty=k/n$}
For fixed positive integer $k$ and $n>k$, the <Fourier coefficient> of the <Fejér kernel> at frequency $k$ multiplies the mode by $1-k/n$. The displayed error is exact. Consequently the order $n^{-1}$ cannot be replaced by $o(n^{-1})$ for all twice continuously differentiable periodic functions, even though each fixed Fourier mode is infinitely differentiable.