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

• Xianghong Chen ^[1] ; Dashan Fan ^[2] ; Juan Zhang
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
  2. [2] niversity of Wisconsin–Milwaukee
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 4, 2020, págs. 1133-1148
• Idioma: inglés
• DOI: 10.4171/rmi/1161
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• Resumen

  • 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. 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.