Most book discuss whether the Fourier series of a function converges. However, I could not easily find many converse statements, that is, when it does converge, what is the property of the limiting function.
Consider the Fourier series $\sum_k c_k e^{2\pi i k}.$ If it converges a.e. to some function, must the function be Lebesgue integrable? What are some results of this kind?