Modified Spherical Harmonics in Several Dimensions

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 $$x_d\Delta u + (d-2)\frac{\partial u }{\partial x_d}= 0$$. 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 $$d=3$$ and $$d=4$$ the modified theory has already been treated.