When I was reading a proof of some problem, it said that "Since $f$ and $f'$ are in $L^2([-\pi,\pi])$, the Fourier Series of $f$ converges to $f$ uniformly".
My question is that, is this statement true?
The version I learned is that, if $f$ is piecewise differentially continuous, then its Fourier Series converges to itself uniformly. But the condition in this version is much stronger than the condition in the statement above, and it makes me feel like the statement can not be true.
Edit: It is true due to the answer posted by Julián Aguirre.
If $\{a_n\}$ is the sequence of Fourier coefficients of $f$, then $$ \sum|a_n|^2<\infty \quad\text{and}\quad\sum n^2|a_n|^2<\infty. $$ Then $$ \sum|a_n|=\sum \frac1n\,(n\,|a_n|)\le\Bigl(\sum\frac{1}{n^2}\Bigr)^{1/2}\Bigl(\sum n^2|a_n|^2\Bigr)^{1/2} <\infty. $$