Modified spherical harmonics in several dimensions
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 \(\mathbb{R}^d\) defines modified spherical harmonics of degree \(n\). Then a non-Euclidean scalar product for functions on the unit half-sphere \(S^{d-1}_+\) is introduced. Modified spherical harmonics of different degrees are found to be orthogonal. This permits to show that the \(n\)th modified zonal harmonic is a reproducing kernel. The addition formula for modified zonal harmonics from an orthonormal basis of modified spherical harmonics of degree \(n\) is shown. It is further shown that the restriction to \(S^{d-1}_+\) in the addition formula can be dropped. Every modified spherical harmonic of degree \(n\) can be obtained from linear combinations of the \(n\)th modified zonal harmonic evaluated at a finite number of points in \(S^{d-1}_+\). The real vector space of all homogeneous polynomials of degree \(n\) in \(\mathbb{R}^d\) can be represented with a Fischer-type decomposition. Section 5 contains several proofs, omitted earlier, followed by references.
generalized axially symmetric potentials
scientific article; zbMATH DE number 7137695
scientific article