Revista Matemática Iberoamericana


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Published online first: 2020-01-07
DOI: 10.4171/rmi/1161

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

Xianghong Chen[1], Dashan Fan[2] and Juan Zhang

(1) Sun Yat-sen University, Guangzhou, China
(2) University of Wisconsin–Milwaukee, USA

We show that there exists an integrable function on the $n$-sphere $(n \geq 2)$, whose Cesàro $(C, (n − 1)/2)$ means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres. We also study relations among Cesàro, Riesz, and Bochner–Riesz means.

Keywords: Cesàro means, Riesz means, spherical harmonic expansion, almost everywhere divergence

Chen Xianghong, Fan Dashan, Zhang Juan: Almost everywhere divergence of spherical harmonic expansions and equivalence of summation methods. Rev. Mat. Iberoam. Electronically published on January 7, 2020. doi: 10.4171/rmi/1161 (to appear in print)