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.