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 LP 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-2 is the celebrated Carleson theorem, proved ill 1966 (and extended to L-p by Hunt in 1967). In this paper, we take the system j(n)(alpha)(X) = root 2(alpha + 2n + 1) J(alpha+2n+1)(X)X-alpha-1, n = 0, 1, 2, ... (with J(n) being the Bessel function of the first kind and of the order mu), which is orthonormal in L-2((0, infinity), X2 alpha+1 dx), and whose Fourier series are the so-called Fourier-Neumann series. We Study the almost everywhere convergence of Fourier-Neumann series for functions in L-p((0, infinity), X2 alpha+1 dx) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established). (C) 2009 Elsevier B.V. All rights reserved.