Contributions to Modified Spherical Harmonics in Four Dimensions

Abstract

A modification of the classical theory of spherical harmonics in four dimensions is presented. The space $\mathbb {R}^4 = \{(x,y,t,s)\}$ is replaced by the upper half space ${\mathbb {R}}_{+}^{4}=\left\{ (x,y,t,s), s > 0 \right\}$, and the unit sphere S in $\mathbb {R}^4$ by the unit half sphere $S_{+}=\left\{ (x,y,t,s): x^2 +y^2+ t^2+ s^2 =1, s > 0 \right\}$. Instead of the Laplace equation $\Delta h = 0$ we shall consider the Weinstein equation $s\Delta u + k \frac{\partial u }{\partial s}= 0$, for $k \in \mathbb {N}$. The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on $S_{+}$. It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct—with respect to this non-Euclidean scalar product—an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation.