2023/12/20 by Annika Moucha, Moucha, Annika
Mathematics · #30F45 #33C20 #35P10 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Primary 30H50 #Secondary 46A35
paper · pdf · doi:10.48550/arxiv.2312.12931
openalex publication_date 2023/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the space H(Ω) of holomorphic functions F:Ω→ℂ, where Ω=\(z,w)∈\widehatℂ2 : z⋅ w≠ 1\, possesses an orthogonal Schauder basis consisting of distinguished eigenfunctions of the canonical Laplacian on Ω. Mapping Ω biholomorphically onto the complex two-sphere, we use the Schauder basis result in order to identify the classical three-dimensional spherical harmonics as restrictions of the elements in H(Ω) to the real two-sphere analogue in Ω. In particular, we show that the zonal harmonics correspond to those functions in H(Ω) that are invariant under automorphisms of Ω induced by Möbius transformations. The proof of the Schauder basis result is based on a curious combinatorial identity which we prove with the help of generalized hypergeometric functions.