More on Modified Spherical Harmonics

Abstract

A modification of the classical theory of spherical harmonics is presented. The space ${\mathbb {R}}^d = \{(x_1,\ldots ,x_d)\}$ is replaced by the upper half space ${{\mathbb {R}}}_{+}^{d}=\left\{ (x_1,\ldots ,x_d), x_d > 0 \right\}$, and the unit sphere $S^{d-1}$ in ${\mathbb {R}}^d$ by the unit half sphere $S_{+}^{d-1}=\left\{ (x_1,\ldots ,x_d): x_1^2 + \cdots + x_d^2 =1, x_d > 0 \right\}$. Instead of the Laplace equation $\Delta h = 0$ we shall consider the Weinstein equation $x_d\Delta u + k \frac{\partial u }{\partial x_d}= 0$, for $k \in {\mathbb {N}}$. The Euclidean scalar product for functions on $S^{d-1}$ will be replaced by a non-Euclidean one for functions on $S_{+}^{d-1}$. It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case $k=d-2$ the modified theory has already been treated by the author.