Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions

What is it about?

This paper presents some generalized FFTs. We start out from an Spherical Gauss-Laguerre (SGL) sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of O(B^4), being the respective bandlimit, while the number of sample points on R^3 scales with B^3. Thisclearly improves the naive bound of O(B^7). At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain.

Featured Image