Convergence of Fourier series

Summary

Let $S_N f$ be the partial sums of the Fourier series of a periodic function $f$. The convergence of Fourier series is the question whether, and in which sense, $S_N f$ tends to $f$. The answer depends on the mode of convergence: at a point, uniformly, almost everywhere, or in the norm of $L^p$. Smoothness or bounded variation of $f$ gives pointwise convergence, while mere continuity does not. In $L^2$ the partial sums always converge, and by Carleson's theorem they also converge almost everywhere for $f \in L^2$.