In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on Rd. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function hκ(x):=∏ν∈R+|x,ν|κν,x∈Rd on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro (C,δ) means in the weighted Lp space for δ above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

We give the sufficient and necessary conditions for the inclusions ΦBV ⊂ ΛBV and ΛBV ⊂ ΦBV. We also give the example such that the above inclusions are not strict.

No abstract

No abstract