Almost everywhere convergence of orthogonal expansions of several variables

For weighted $L^1$ space on the unit sphere of ${\text{RR}}^{d+1}$, 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 ${\text{RR}}^d$.