This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!


Suppose $f(x)=\sum\_{k\in\mathbb{Z}}c\_ke^{ikx}$, whereby we mean pointwise convergence. What properties must $f(x)$ then satisfy? Clearly continuity is too strong (take for example an appropriately defined square wave). $L^1[-\pi,\pi]$ seems troublesome as well, since term-by-term integration is not necessarily valid with only pointwise convergence. 


Thanks ahead for any tips!



Weakest assumption for pointwise convergence of Fourier series
The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting [survey paper](http://dx.doi.org/10.1090/S0002-9904-1955-09853-7) on the subject:



> 
> One of the problems in the theory of trigonometric series
>  $$\frac12a\_0+\sum\_{n=1}^\infty(a\_n\cos nx+b\_n\sin nx)\tag{1.1}$$ is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function $f(x)$, then $f(x)$ is necessarily integrable and the coefficients, $a\_n$ and $b\_n$, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (...) The problem has been solved by Denjoy [4; 5], Verblunsky [14],
>  Marcinkiewicz and Zygmund [10], Burkill [1; 2], and James [8]. (...) The solutions are described, mainly in the order in which they were published, in §§2-7 below.
> 
> 
> 


