Consider the Fourier series (in exponential form) generated by a function $f$ which is continuous on $[0,2\pi]$ and periodic with period $2\pi$ , say
$f(x)\sim\sum_{n = - \infty }^{+ \infty }\alpha _{n}e^{inx}.$
Assume also that the derivative f '\in \mathbb{R} on $[0,2\pi]$.
a.) Prove that the series $\sum_{n = - \infty }^{+ \infty }n^{2}\left | \alpha _{n} \right |^{2}$ converges; then use the Cauchy-Schwarz inequality to deduce that $\sum_{n = - \infty }^{+ \infty }\left | \alpha _{n} \right |$ converges.
b.) From (a), deduce that the series $\sum_{n = - \infty }^{+ \infty }\alpha _{n}e^{inx}$ converges uniformly to a continuous sum function $g$ on $[0,2\pi]$. Then prove that $f=g$.