Just a few comments about convergence. In general square integrability will not guarantee convergence of the Fourier series to the original function. For example, in the case of the sawtooth mentioned above, the Fourier series will take on the value 0 at x= L and x= -L, both of which are points of discontinuity.
Convergence of the Fourier series is rather unlike convergence of a power series. A power series converges in a disk. In contrast the convergence of a fourier series is a local phenomenon. So, whether or not the fourier series converges depends on the local properties of the function at that point. In particular a sufficient condition for the fourier series to converge to the original function is that the function satisfy a Lipshitz condition at that point. This means that abs(f(x)-f(y)) can be made smaller that const*abs(x-y) in some interval about x.(Essentially a Lipshitz condition is stronger than continuity but weaker than differentiability).
A closely related subject is that of Fourier transforms.