The surprising almost everywhere convergence of Fourier–Neumann series

Abstract

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L p 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 in 1966 (and extended to L p by Hunt in 1967). In this paper, we take the system j n α ( x ) = 2 ( α + 2 n + 1 ) J α + 2 n + 1 ( x ) x − α − 1 , n = 0 , 1 , 2 , … (with J μ being the Bessel function of the first kind and of the order μ ), which is orthonormal in L 2 ( ( 0 , ∞ ) , x 2 α + 1 d x ) , 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 , ∞ ) , x 2 α + 1 d x ) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established).