Let $f$ be a smooth $2\pi$-periodic function, i.e. a smooth function on the torus $\TT$, $A_f$ the convolution operator $$ A_f\psi(x) \colon=\frac1{2\pi}\int_{-\pi}^\pi f(x-y)\psi(y)\,dy =\frac1{2\pi}\int_{-\pi}^\pi\psi(x-y)f(y)\,dy $$ and $P$ the differential operator $P\psi\colon=-i\psi^\prime$ for every smooth $2\pi$-periodic function $\psi$. Prove the following assertions: 1. $A_f$ and $P$ commute. 2. For all $n\in\Z$ $\psi_n(x)=e^{inx}$ is an eigen-function of $P$. 3. $\psi_n$ is an eigen-function of $A_f$ and the corresponding eigen-value is $\wh f(n)=(2\pi)^{-1}\int_{-\pi}^\pi e^{-iny}f(y)\,dy$, i.e. the $n$-th Fourier coefficient of $f$. 4. For all $2\pi$-periodic smooth functions $\vp,\psi$: $\la P\psi,\vp\ra=\la\psi,P\vp\ra$.

2. Since $\psi$ is smooth all its derivatives on the compact space $\TT$ are bounded. Thus, by a result on parameter integrals: $$ (PA_f)\psi(x) =\frac{-i}{2\pi}\pa_x\int_{-\pi}^\pi\psi(x-y)f(y)\,dy =\frac{-i}{2\pi}\int_{-\pi}^\pi\psi^\prime(x-y)f(y)\,dy =A_f(P\psi)(x)~. $$ 3. $P\psi_n(x)=-i(in)e^{inx}=n\psi_n(x)$ and since $\psi_n(x-y)=\psi_n(x)\psi_n(-y)=\psi_n(x)e^{-iny}$, we get $$ A_f\psi_n(x) =\frac1{2\pi}\int_{-\pi}^\pi\psi_n(x-y)f(y)\,dy =\psi_n(x)\frac1{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy =\wh f(n)\psi_n(x)~. $$ 4. By integration by parts and $2\pi$-periodicity of $\psi\bar\vp$ we get \begin{eqnarray*} \la P\psi,\vp\ra &=&\frac{-i}{2\pi}\int_{-\pi}^\pi\psi^\prime(x)\bar\vp(x)\,dx\\ &=&\frac{-i}{2\pi}\Big(\psi(x)\bar\vp(x)|_{-\pi}^\pi-\int_{-\pi}^\pi\psi(x)\bar\vp^\prime(x)\,dx\Big)\\ &=&\frac{i}{2\pi}\int_{-\pi}^\pi\psi(x)\bar\vp^\prime(x)\,dx =\la\psi,P\vp\ra~. \end{eqnarray*}