We give a sufficient condition for a zonal function to be a strictly positive definite. A major result of this thesis is that we have proven Shreiner's result as a consequence of a more general representation-theoretic result, namely for compact groups using tools from the Representation Theory.

The other major direction of this thesis is to compress square integrable functions on the unit sphere. The main tools used in this analysis are a Ramanujan set of rotations and plan wavelets.

This dissertation presents contributions to research in the field of mathematics and digital signal processing on the sphere. The main goal is two fold: (i) Study interpolation, equidistribution and covering on the sphere using mathematical tools such as the representation theory. (ii) Develop an algorithm using planar wavelets and the Ramanujan set of rotations SM5 to compress functions on the unit sphere.