A non-linear approximation method on the sphere

We show the applicability of a modified version of the recently developed Regularized Functional Matching Pursuit (RFMP) to the approximation of functions on the sphere from grid-based data. We elaborate the mathematical details of the choice of trial functions and the specifics of the algorithm. Moreover, we show numerical examples for some benchmarks. The dictionary of trial functions contains orthogonal polynomials (spherical harmonics) as well as spherical scaling functions and wavelets. It turns out that the greedy algorithm RFMP yields sparse approximations by combining different types of trial functions in a (particular) optimal way, where the sparsity can essentially be increased by a-priori choosing the dictionary appropriately. Moreover, the result of the RFMP can be used for a multiresolution analysis of the investigated function. © 2014, Springer-Verlag Berlin Heidelberg.