$f$ is a $2\pi$ periodic continuously differentiable function on $[-\pi,\pi]$

Question

The question :

Let $f$ be a $2\pi$ in $C^1$ function on $[-\pi,\pi].$ Let $c_n$ be the Fourier coefficient of $f: c_n := \hat{f}(n)= \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{inx}dx$. Show that $\sum_{-\infty}^{\infty} n^2|c_n|^2<\infty $

I thought I can solve this problem by using Parseval's theorem, but I failed. Can you help me?

Answer 1

Hint: Integrate by parts to obtain a formula for $(f')\hat {} (n)$ in terms of $\hat f(n).$