INTEGRATION OF THE INTERTWINING OPERATOR FOR h-HARMONIC POLYNOMIALS ASSOCIATED TO REFLECTION GROUPS

Let V be the intertwining operator with respect to the reflection invariant measure hαdω on the unit sphere S d−1 in Dunkl’s theory on spherical h-harmonics associated with reflection groups. Although a closed form of V is unknown in general, we prove that ∫ Sd−1 V f(y)hα(y)dω = Aα ∫ Bd f(x)(1 − |x|2)|α|1−1dx, where Bd is the unit ball of Rd and Aα is a constant. The result is used to show that the expansion of a continuous function as Fourier series in h-harmonics with respect to hαdω is uniformly Cesáro (C, δ) summable on the sphere if δ > |α|1 + (d− 2)/2, provided that the intertwining operator is positive.