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

TL;DR

This paper demonstrates that for certain integrable functions on high-dimensional spheres, common summation methods for spherical harmonic expansions diverge almost everywhere, extending classical results and establishing equivalences among different summation techniques.

Contribution

It shows the divergence of Cesàro, Riesz, and Bochner-Riesz means for specific functions on spheres, extending Stein's and Taibleson's results to higher dimensions.

Findings

Cesàro means diverge almost everywhere for some functions

Results extend classical divergence theorems to spheres of dimension n≥2

Establish equivalence of divergence among different summation methods

Abstract

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Ces\`aro (C,$\frac{n-1}{2}$) 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.