Let $f$ be a $2\pi$ periodic function on $[0,2\pi]$. If $f$ is absolutely continuous, is it true that the sum of its Fourier coefficient converges absolutely $ S(t)=\sum_{n=-\infty}^{\infty} | \hat f (n) | < \infty \text{ ?} $ If so how do we proove it.
My understanding, which I think is wrong, is that the derivative $f'$ is in $L^2$ (how do we justify that?). But then how do I bound $S(t)$?