Fast spherical Fourier algorithms

Spherical Fourier series play an important role in many applications. A numerically stable fast transform analogous to the fast Fourier transform is of great interest. For a standard grid of O(N2) points on the sphere, a direct calculation has computational complexity of O(N4), but a simple separation of variables reduces the complexity to O(N3). Here we improve well-known fast algorithms for the discrete spherical Fourier transform with a computational complexity of O(N2 log2 N). Furthermore we present, for the first time, a fast algorithm for scattered data on the sphere. For arbitrary O(N2) points on the sphere, a direct calculation has a computational complexity of O(N4), but we present an approximate algorithm with a computational complexity of O(N2 log2 N).