Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 1 / d / Solution

Write the Fourier coefficients as $c_k=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }u(t)e^{-ikt}dt$. The partial sum through frequencies $-n,\dots ,n$ is therefore a convolution withThe last expression follows by summing a finite geometric series, with its removable value $2n+1$ at zero. Periodicity and the evenness of $D_n$ giveThis also proves linearity and that the image lies in $F_n$. We use the continuous periodic function space, consistent with the premise and the displayed maximum norm.

The convolution formula gives the upper boundTo prove sharpness within continuous functions, takeIt is continuous and periodic, with norm at most one. At $x=0$, dominated convergence gives $(S_n{u}_{ϵ})(0)\to (2\pi {)}^{-1}\int |D_n|$. Hence that upper bound is the actual operator norm. Evenness yieldsThe approximating continuous signs avoid assuming that the discontinuous sign of the kernel belongs to the domain space.