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Modified Spherical Harmonics in Several Dimensions

  • Heinz LeutwilerEmail author
Article
  • 34 Downloads

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.

Keyword

Spherical harmonics, Generalized axially symmetric potentials, Generalized function theory 

Mathematics Subject Classification

30G35 33A45 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFriedrich-Alexander-University Erlangen-NurembergErlangenGermany

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