The name is also often used to refer to the extension of the result by Richard Hunt (1968) to Lp functions for p ∈ (1, ∞] (also known as the Carleson–Hunt theorem) and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods.
By strengthening the continuity assumption slightly one can easily show that the Fourier series converges everywhere.
This was proven by Dirichlet, who expressed his belief that he would soon be able to extend his result to cover all continuous functions.
This was disproved by Paul du Bois-Reymond, who showed in 1876 that there is a continuous function whose Fourier series diverges at one point.
Before Carleson's result, the best known estimate for the partial sums sn of the Fourier series of a function in Lp was
In other words, the function sn(x) can still grow to infinity at any given point x as one takes more and more terms of the Fourier series into account, though the growth must be quite slow (slower than the logarithm of n to a small power).
This result had not been improved for several decades, leading some experts to suspect that it was the best possible and that Luzin's conjecture was false.
Carleson said in an interview with Raussen & Skau (2007) that he started by trying to find a continuous counterexample and at one point thought he had a method that would construct one, but realized eventually that his approach could not work.
Expositions of the original paper Carleson (1966) include Kahane (1995), Mozzochi (1971), Jørsboe & Mejlbro (1982), and Arias de Reyna (2002).
Charles Fefferman (1973) published a new proof of Hunt's extension which proceeded by bounding a maximal operator.
Konyagin (2000) improved Kolmogorov's counterexample by finding functions with everywhere-divergent Fourier series in a space slightly larger than Llog+(L)1/2.
One can ask if there is in some sense a largest natural space of functions whose Fourier series converge almost everywhere.