Fractal Spherical Harmonics

This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.