Let $f$ be a $2\pi$ periodic function integrable on $[-\pi,\pi]$ with $f'$ integrable on $[-\pi,\pi]$. Show that $\sum^{\infty}_{n=-\infty}|\hat f(n)|<\infty$
I know $(in)\hat f(n)=\hat {f'}(n)$, but it seems to tell nothing about the infinite sum, also I know $\sum^{\infty}_{-\infty}|\hat f(n)|^2=\frac{1}{2\pi}\int^{\pi}_{-\pi}|f(x)|^2dx<\infty$ by Parseval's identity, but it has a extra square and I have no idea how to remove that square to get the desired results. Could you please give me some hints?