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

Abstract

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, 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.