Almost everywhere convergence of orthogonal expansions of several variables

Abstract

For weighted $L^1$ space on the unit sphere of \RR^{d+1}, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in $h$-harmonics. The result applies to various methods of summability, including the de la Vall\'ee Poussin means and the Ces\`aro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of \RR^d.Comment: 23 page