Spherical harmonic analysis and synthesis for global multiresolution applications

On the Earth and in its neighborhood, spherical harmonic analysis and synthesis are standard mathematical procedures for scalar, vector and tensor fields. However, with the advent of multiresolution applications, additional considerations about convolution filtering with decimation and dilation are required. As global applications often imply discrete observations on regular grids, computational challenges arise and conflicting claims about spherical harmonic transforms have recently appeared in the literature. Following an overview of general multiresolution analysis and synthesis, spherical harmonic transforms are discussed for discrete global computations. For the necessary multirate filtering operations, spherical convolutions along with decimations and dilations are discussed, with practical examples of applications. Concluding remarks are then included for general applications, with some discussion of the computational complexity involved and the ongoing investigations in research centers.