Mean and almost everywhere convergence of Fourier-Neumann series

Let J(mu) denote the Bessel function of order mu. The functions x(-alpha/2-beta/2-1/2)J(alpha+beta+2n+1)(x(1/2)), n = 0, 1,2,..., form an orthogonal system in L-2((0, infinity), x(alpha+beta)dx) when alpha + beta > -1. In this paper we analyze the range of p, alpha, and beta for which the Fourier series with respect to this system converges in the L-p((0, infinity), x(alpha) dx)-norm. Also, we describe the space in which the span of the system is dense and we show some of its properties. Finally, we study the almost everywhere convergence of the Fourier series for functions in such spaces. (C) 1999 Academic Press.