What are the conditions for existence of the Fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$ What are the conditions for existence of the Fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$?

In addition to Carleson's theorem (stated by AD above), which gives a sufficient condition for pointwise convergence almost everywhere, one might also consider the following theorem about uniform convergence:

Suppose $f$ is periodic. Then, if $f$ is $\mathcal{C}^0$ and piecewise $\mathcal{C}^1$, $S_N(f)$ converges uniformly to $f$ on $\mathbb{R}$.

