h-Harmonics and Analysis on the Sphere

Abstract

Dunkl h-harmonics are defined as homogeneous polynomials satisfying the Dunkl Laplacian equation. They are defined and studied in Section 3.1. Projection operators and orthogonal expansions in spherical h-harmonics are studied in Section 3.2, which includes a concise expression for the reproducing kernel of the spherical h-harmonics. This expression is an analog of the zonal harmonics, which suggests a definition of a convolution operator, defined in Section 3.3 and it helps us to study various summability methods for spherical h-harmonic expansions.