Gegenbauer kernel filtration on the unit hypersphere

TL;DR

This paper derives explicit convolution formulas for Gegenbauer kernel filtration on the unit hypersphere, addressing challenges in higher-dimensional manifold smoothing and providing a theoretical foundation for improved visualization techniques.

Contribution

It introduces explicit convolution formulas for Gegenbauer kernel filtration on the unit hypersphere and proves its relation to hyperspherical Legendre harmonics.

Findings

Derived explicit convolution formulas for Gegenbauer kernel filtration.

Proved the limit relation of Gegenbauer filtration to hyperspherical Legendre harmonics.

Addressed higher-dimensional manifold smoothing challenges.

Abstract

Filtration of quantifiable objects by smoothing kernels on Riemannian manifolds for visualisation is an ongoing research. However, using common filters created for linear domains on manifolds with non-Euclidean topologies can yield misleading results. While there is a lot of ongoing research on convolution of quantifiable functions with smoothing kernels on the lower dimensional manifolds, higher-dimensional problems particularly pose a challenge. One important generalization of lower dimensional compact Riemannian manifolds is the unit hypersphere. In this paper, we derive explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of unit hypersphere. We prove that the Gegebauer filtration is the limit of a sequence of finite linear combinations of the hyperspherical Legendre harmonics, among other results.