On azimuthally symmetric 2-sphere convolution

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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.