Convergence of Fourier Series - Convergence Almost Everywhere

Convergence Almost Everywhere

The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s and remained open until finally resolved positively in 1966 by Lennart Carleson. Indeed, Carleson showed that the Fourier expansion of any function in L2 converges almost everywhere. This result is now known as Carleson's theorem. Later on Richard Hunt generalized this to Lp for any p > 1. Despite a number of attempts at simplifying the proof, it is still one of the most difficult results in analysis.

Contrariwise, Andrey Kolmogorov, in his very first paper published when he was 21, constructed an example of a function in L1 whose Fourier series diverges almost everywhere (later improved to divergence everywhere).

It might be interesting to note that Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a continuous function ƒ such that the Fourier series of ƒ fails to converge on any point of E.