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 LpLp 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 L2L2 is the celebrated Carleson theorem, proved in 1966 (and extended to LpLp by Hunt in 1967).In this paper, we take the system jnα(x)=2(α+2n+1)Jα+2n+1(x)x−α−1,n=0,1,2,… (with JμJμ being the Bessel function of the first kind and of the order μμ), which is orthonormal in L2((0,∞),x2α+1dx), and whose Fourier series are the so-called Fourier–Neumann series. We study the almost everywhere convergence of Fourier–Neumann series for functions in Lp((0,∞),x2α+1dx) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established).