Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L n−l−1(l+1∕2)​(r2)r l Y lm (ϑ, φ), \(\vert m\vert \leq l < n \in \mathbb{N}\), L n−l−1(l+1∕2)​ being a generalized Laguerre polynomial, Y lm a spherical harmonic, constitute an orthonormal basis of the space L2​ on \(\mathbb{R}^{3\!}\) with Gaussian weight exp(−r2). 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 \(\mathscr{O}(B^{4})\), B being the respective bandlimit, while the number of sample points on \(\mathbb{R}^{3\!}\) scales with B3​. This clearly improves the naive bound of \(\mathscr{O}(B^{7})\). 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 \(\mathscr{O}(B^{3}\log ^{2\!}B)\) fast SGL Fourier transform and inverse, and an outlook on future developments.