Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function Pi(j=1)(d) \x(j)\2(kappaj) (kappa(j)greater than or equal to0) on the unit sphere Sd-1 is studied. The main result characterizes the critical index of summability of the Cesaro (C, delta) means of the h-harmonic expansion; it is proved that the (C, 6) means of any continuous function converge uniformly in the norm of C(Sd-1) if and only if delta > (d - 2)/2 + Sigma(j=1)(d) kappaj - min(1less than or equal tojless than or equal tod) kappa(j). Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd-1, the (C, delta) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if delta > (d - 2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions Pi(j=1)(d) \x(j)\(2kappaj) (1-\x\(2))(mu-1/2) don the unit ball B-d and Pi(j=1)(d) x(j)(kappaj-1/2) (1 - \x\(1))(mu-1/2) on the simplex T-d. As a related result, the Cesdro summability of the generalized Gegenbauer expansions associated to the weight function \t\(2mu)(1 - t(2))(lambda-1/2) on [- 1: 1] is Studied, which is of interest in itself. (C) 2003 Elsevier Science (USA). All rights reserved.