After elaborating the integral that now bears his name, Henri Lebesgue has been studying the convergence of Fourier series in the early XXth century. In a series of articles, among which one is published in 1909  in the Annales de la Faculté des Sciences de Toulouse, Lebesgue analyzes approximations of unity (in modern language). He obtains several uniform convergence results of Fourier series, for instance for Lipschitz functions or more generally for Dini-Lipschitz functions, but he also obtains divergence results : Fourier series of a continuous function that does not converge everywhere (the first example is due to du Bois-Reymond), or Fourier series of a continuous function that converges everywhere but not uniformly.

Lebesgue's work was later extended by several striking results related to the convergence of Fourier series, the subject being still very active.