Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems

Abstract

We study some problems related to convergence and divergence a.e. for Fourier series in systems { ϕ k } \{ {\phi _k}\} , where { ϕ k } \{ {\phi _k}\} is either a system of orthonormal polynomials with respect to a measure d μ d\mu on [ − 1 , 1 ] [-1,1] or a Bessel system on [ 0 , 1 ] [0,1] . We obtain boundedness in weighted L p {L^p} spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on [ − 1 , 1 ] [-1,1] .