On azimuthally symmetric 2-sphere convolution

We consider the problem of azimuthally symmetric convolution of signals defined on the 2-Sphere. Applications of such convolution include but are not limited to: geodesy, astronomical data (such as the famous Wilkinson Microwave Anisotropy Probe data), and 3D beamforming/sensing. We review various definitions of convolution from the literature and show a nontrivial equivalence between different definitions. Some convolution formulations based on SO(3) are shown not to be well formed for applications and we demonstrate a simpler framework to understand, use and generalize azimuthally symmetric convolution.