Let $f\in L^2([-1/2,1/2])$.
Define $a_n=\int_{[-1/2,1/2]} f(x) e^{-2\pi i n x} dx$ for each $n\in\mathbb{Z}$.
Define $S_N(x)=\sum_{n=-N}^N a_n e^{2\pi i n x}$ for each $N\in \mathbb{Z}^+$ and $x\in \mathbb{R}$.
I know that $S_N\rightarrow f$ in $L^2$-norm, but I'm curious whether $S_N\rightarrow f$ pointwise a.e.. What more conditions should one assume to assert that $S_N\rightarrow f$ pointwise a.e.? How do I prove it?
The general statement is a very difficult problem solved in the 20th century, the result is called Carleson's theorem. It says in particular that the Fourier series of a $L^2$ periodic function converges to it a.e.
Weaker theorems, e.g. by assuming the function is continuous and bounded variation, are much easier to prove.