Past exam of the fourier analysis course of the University of Oxford / 2023 / 3 / a / iii

Let $D={\sum }_{k\in \mathbb{Z}}{\delta }_{2\pi k}$ be the Dirac comb. Pair the given identity $D=(2\pi {)}^{-1}{\sum }_{n\in \mathbb{Z}}{e}^{in\cdot }$ with the Schwartz function $y\mapsto \varphi (x-y)$. On the left this gives ${\sum }_k\varphi (x-2\pi k)=P\varphi (x)$. On the right,Therefore the Fourier series isThe Fourier bounds for Schwartz functions imply $|\hat{\varphi }(n)|\le C_N(1+|n|{)}^{-N}$ for every $N$. For each nonnegative integer $j$, choose $N>j+1$ to obtain ${\sum }_n|n{|}^j|\hat{\varphi }(n)|<\infty$. Thus the Fourier series and every differentiated Fourier series converge absolutely and uniformly on all of $\mathbb{R}$. In particular convergence holds in the topology of smooth $2\pi$ periodic functions, defined by uniform convergence of every derivative on one period, and also in tempered distributions.