Let us say we have a function $f[a,b]\rightarrow \mathbb{C}$ and we want to find the Fourier series for this function in the interval $[a,b]$ what is the condition on $f$ such that the Fourier series will converge to the function in this range?
My thoughts
I am guessing that the only requirement be that $f\in L^2(a,b)$ but I don't know if this is correct or if it is the reasoning behind it.
On
It's a very deep, celebrated theorem by Carleson (1966) saying that the Fourier series of an $L^2$ function converges pointwise almost everywhere.
On the other hand there are examples (already by Kolmogorov in 1876) of $L^1$ functions whose Fourier series diverge everywhere.
More elementary results are, for example that the Fourier series of a continuous, piecewise differentiable function converges pointwise everywhere. See this for a thorough discussion about convergence of Fourier series.
On
If the function is in $L^2[a,b]$, then the Fourier series for $f$ will converge in the $L^2$ norm to $f$. More remarkably, the Fourier series will converge pointwise a.e. to $f$, which is a result of Lennart Carleson known as Carleson's Theorem. This is considered to be one of the deepest results of Fourier Analysis. There are no known simple proofs.
Fourier Series won't converge point-wise but on average.
From the start Fourier transform is (usually) defined on the Schwarz class of infinitely differentiable functions of compact support (which is a bit of a requirement), but it can be shown that functions in $L^2$ can be approximated arbitrarily well by functions which are Fourier transformable. I have seen one of the proofs of this in a course so it should probably be readily available online. The version I saw used the derivatives of $e^{-x^2}$ to build a basis and then showed that it was complete in $L^2$ and also Fourier transformable. Here seems to be some details on that. In this linked version they are not only derivatives but also renormalized to have integral 1 "probabilists" (probability distribution sums to 1). Probably out of convenience to make it an ON basis.