Modified Spherical Harmonics

Abstract

A modification of the classical theory of spherical harmonics is presented. The unit sphere S in $${\mathbb{R}^3 = \{(x,y,t)\}}$$ is replaced by the half-sphere S + in the upper half space, the Euclidean scalar product on S by a non-Euclidean one on S +, and the Laplace equation $${\Delta h = 0}$$ by the equation $${t\Delta v + \frac{\partial v }{\partial t}= 0}$$ . It will be shown that most results from the theory of spherical harmonics in $${\mathbb{R}^3}$$ stay valid in this modified setting.