Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

Mathematics

Han Feng¹  Yan Ge¹ 

[1] Department of Mathematics, City University of Hong Kong, Hong Kong, China;

关键词: spherical h-harmonics;    Cesàro means;    Christoffel functions;   

DOI  :  10.3390/math10122108

来源: DOAJ

【 摘 要 】

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 ${\mathbb{R}}^d$. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function $h_{\kappa }\left(x\right):={\prod }_{\nu \in R+}{\left|⟨x,\nu ⟩\right|}^{{\kappa }_{\nu }},x\in {\mathbb{R}}^d$ 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 $(C,\delta )$ means in the weighted $L^p$ space for $\delta$ above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

【 授权许可】

Unknown