  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.
