TITLE: Weakest assumption for pointwise convergence of Fourier series
QUESTION [5 upvotes]: 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!

REPLY [7 votes]: The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting survey paper 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.