Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions

Abstract

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L (l+1/2)n−l−1 being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L 2 on R 3 with Gaussian weight exp(−r 2 ). These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an 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 ), B being the respective bandlimit, while the number of sample points on R 3 scales with B 3 . This clearly 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. We conclude with a discussion, including the layout of a true O(B 3 log 2 B) fast SGL Fourier transform and inverse, and an outlook on future developments.

Translation matrix elements for spherical Gauss–Laguerre basis functions

“…This approach has the major advantage that the SGL Fourier coefficients of f and g need to be computed only once for evaluating the overlap integral I for several different rotations and translations. For exactly this purpose, we have recently described fast Fourier transforms for the SGL basis functions -see [Prestin and Wülker, 2017] for gridded data, and [Wülker, 2018, Chap. 3] for scattered data.…”

“…2.1) constitute an orthonormal polynomial basis of the space L 2 on R 3 equipped with the radial Gaussian weight function exp(−r 2 ). We have recently described reliable fast Fourier transforms for the SGL basis functions [Prestin and Wülker, 2017]. These algorithms allow for a fast computation of SGL Fourier coefficients for spectral analysis.…”

Fast SGL Fourier transforms for scattered data

“…In particular, a function f ∈ H is called bandlimited with bandwidth B if the SGL Fourier coefficientsf nlm := f, H nlm H vanish for n > B. We have recently described fast and reliable SGL Fourier transforms, i. e., generalized FFTs for the SGL basis functions [Prestin and Wülker, 2017]. These fast algorithms compute the B(B + 1)(2B + 1)/6 potentially non-zero SGL Fourier coefficientsf nlm of a function f with bandwidth B in O(B 4 ) or even only O(B 3 log 2 B) computation steps from (2B) 3 sampled function values, instead of the naive O(B 6 ) computation steps (as usual, we define a single computation step as a complex multiplication and subsequent addition).…”