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 Vallée Poussin means and the Cesàro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of $\RR^d$.