Fejér second-modulus approximation bound ($\Vert {\sigma }_{n}f-f{\Vert }_{\infty }\le (1+{\pi }^2){\omega }_2(f,n^{-1/2})$)

The even positive Fejér kernel of total mass one gives ${\sigma }_{n}f-f=\frac{1}{2}\int F_n(t)[f(\cdot +t)-2f+f(\cdot -t)]dt$. The scaling property of the second modulus of smoothness bounds this by ${\omega }_2(f,\delta )(1+{\delta }^{-2}\int t^2{F}_n(t)dt)$. On $[-\pi ,\pi ]$, $\sin (|t|/2)\ge |t|/\pi$ gives $\int t^2{F}_n(t)dt\le {\pi }^2/n$. Choose $\delta =n^{-1/2}$. For twice continuously differentiable functions, the second-difference integral formula makes the error $O(n^{-1})$.

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.