New fractal functions on the sphere

Abstract

In this article, a family of continuous functions on the unit sphere \(S\subseteq {\mathbb {R}}^{3}\) is considered as a generalization of spherical harmonics. The family is fractalized using a linear and bounded operator of functions on the sphere. Particular values of the scale vector in the iterated function system (IFS) may yield classical functions system on the sphere. We have shown that for different values of the scale vector in the IFS, Bessel sequences, frames, and Riesz bases can be established for the space \({\mathcal {L}}^{2}(S)\) of square integrable functions on the sphere.