Mean and Almost Everywhere Convergence of Fourier]Neumann Series

Let J denote the Bessel function of order m. The functions m xya r2yb r2y1r2J x1r2 ., ns0, 1, 2, . . . , form an orthogonal system in aqbq2 nq1 L2 0, `.,xaqbdx. when a qb )y1. In this paper we analyze the range of p, a, and b for which the Fourier series with respect to this system converges in the L p 0, `., xa 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.