Mean and Almost Everywhere Convergence of Fourier–Neumann Series

AbstractLet Jμ denote the Bessel function of order μ. The functions x−α/2−β/2−1/2Jα+β+2n+1(x1/2), n=0,1,2,…, form an orthogonal system in L2((0,∞),xα+βdx) when α+β>−1. In this paper we analyze the range of p, α, and β for which the Fourier series with respect to this system converges in the Lp((0,∞),xαdx)-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