Spectral synthesis of the invariant Laplacian and complexified spherical harmonics

Abstract

We show that the space $\mathcal{H}(\Omega )$ of holomorphic functions $F:\Omega \to \mathbb{C}$, where $\Omega =\{(z,w)\in {\hat{\mathbb{C}}}^2:z\cdot w\ne 1\}$, possesses an orthogonal Schauder basis consisting of distinguished eigenfunctions of the canonical Laplacian on $\Omega$. Mapping $\Omega$ 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 $\mathcal{H}(\Omega )$ to the real two-sphere analogue in $\Omega$. In particular, we show that the zonal harmonics correspond to those functions in $\mathcal{H}(\Omega )$ that are invariant under automorphisms of $\Omega$ 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.

Cited by in corpus (1)

• Function Theory off the complexified unit circle: Fréchet space structure and automorphisms