Modified spherical harmonics in several dimensions
type: Item
instance of: scholarly article
title: Modified spherical harmonics in several dimensions
zbMATH Open document ID: 1446.30079
author: Heinz Leutwiler
published in: Advances in Applied Clifford Algebras
publication date: 2019-11-27T00:00:00Z
review text: This paper is written for researchers in higher-dimensional harmonic function theory in the framework of modified Clifford analysis, which may also be viewed as a higher-dimensional generalization of modified quaternionic analysis of \(\mathbb{R}^3\) and \(\mathbb{R}^4\). Section 1: Essentially solutions to Weinstein's equation are considered with \(\lambda = d-2\), where \(d\) is the dimension of the embedding Euclidean space \(\mathbb{R}^d\) for the unit half spheres \(S^{d-1}_+\), to which the solutions of Weinstein's equation are to be restricted. The solutions are called modified harmonic functions, instead of generalized axially symmetric potentials. With \(\lambda = d-2\), the Weinstein equation can also be viewed as a certain type of Laplace-Beltrami equation. Homogeneous modified harmonic polynomials of modified Clifford analysis are defined. Section 2 studies several properties of modified harmonic functions. For example, the non-empty intersection of an open connected set domain with the hyperplane \(x_d=0\) uniquely determines the modified harmonic function on the open connected set domain itself. This results in an even symmetry in the \(x_d\) coordinate. Two special singular solutions are stated. The \(\mathbb{R}\)-vector space of all homogeneous modified harmonic polynomials of degree \(n\) on \(\mathbb{R}^d\) is given in terms of homogeneous modified harmonic polynomials. Its dimension is given. A modified Kelvin transform of modified harmonic functions is introduced, the latter are obtained from partial derivatives of the function \(1/r^{2d-4}\). In Section 3 a Poisson-type integral for modified harmonic functions is studied. A polynomial analogue of classical zonal harmonics is introduced, called modified zonal harmonics. They are applied to give an expansion of the Poisson-type integral kernel. In Section 4, a restriction to the half sphere \(S^{d-1}_+\) of the vector space of homogeneous modified harmonic polynomials of degree \(n\) on \(\mathbb{…
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 7137695
zbMATH Keywords: spherical harmonics
zbMATH Keywords: generalized axially symmetric potentials
zbMATH Keywords: generalized function theory
zbMATH Keywords: Clifford analysis
zbMATH Keywords: Weinstein's equation
MaRDI profile type: Publication
full work available at URL: s00006-019-1021-9
OpenAlex ID: W2980596013
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Wikidata QID: Q127026232
DOI: 10.1007/S00006-019-1021-9
description: scientific article; zbMATH DE number 7137695