The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L is the celebrated Carleson theorem, proved in 1966 (and extended to L by Hunt in 1967). In this paper, we take the system j n (x) = √ 2(α+ 2n+ 1) Jα+2n+1(x)x , n = 0, 1, 2, . . . (with Jμ being the Bessel function of the first kind and of the order μ), which is orthonormal in L((0,∞), x dx), and whose Fourier series are the so-called Fourier-Neumann series. We study the almost everywhere convergence of FourierNeumann series for functions in L((0,∞), x dx) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established).