Analysis of Gegenbauer kernel filtration on the hypersphere

Omenyi Et Al

doi:10.21833/ijaas.2021.11.001

Abstract

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Highlights

The role of classical orthogonal polynomials such as the Gegenbauer polynomials as reproducing kernels for the spaces of spherical harmonics of a given degree, or more generally, as providing an explicit construction of symmetry adapted basis functions for those spaces has been studied extensively (Camporesi, 1990; Bezubik and Strasburger, 2006; Omenyi and Uchenna, 2019)
The goal of the present paper is to present a novel form of the Gegenbauer kernel filtration of harmonic functions on the hypersphere
Sn−1 is a linear combination of spherical harmonics of the order less than or equal to l

Summary

Introduction

The role of classical orthogonal polynomials such as the Gegenbauer polynomials as reproducing kernels for the spaces of spherical harmonics of a given degree, or more generally, as providing an explicit construction of symmetry adapted basis functions for those spaces has been studied extensively (Camporesi, 1990; Bezubik and Strasburger, 2006; Omenyi and Uchenna, 2019). The goal of the present paper is to present a novel form of the Gegenbauer kernel filtration of harmonic functions on the hypersphere. It is known, in general, that there is no explicit expression for the fundamental solution of a Laplace-type operator on a Riemannian manifold (Aubin, 1998; Cohl and Palmer, 2015). We aim to demonstrate that with the Gegenbauer filtration kernel, a closedform of fundamental solution can be constructed. This puts in limelight signal processing methods on non-Euclidean spaces and in particular on the hypersphere. We derive some general formulae for the Gegenbauer filtration of functions on Sn, including recent generalizations of Fourier spherical harmonic expansions and discuss their function theoretic consequences

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Conclusion