Almost everywhere divergence of spherical harmonic expansions and equivalence of summation methods

2018/06/10 by Chen, Xianghong, Fan, Dashan, Zhang, Juan
#33C55 #40G05 #43A50 #43A85 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.03727

Abstract

We show that there exists an integrable function on the n-sphere (n≥ 2), whose Cesàro (C,(n-1)/(2)) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres.

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