  Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of
the type $L_{n-l-1}^{(l + 1/2)} (r^2) r^{l} Y_{lm}(\vartheta,\varphi)$, $|m|
\leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized
Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal
basis of the space $L^{2}$ on $\mathbb{R}^{3}$ 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 $\mathcal{O}(B^{4})$, $B$
being the respective bandlimit, while the number of sample points on
$\mathbb{R}^{3}$ scales with $B^{3}$. This clearly improves the naive bound of
$\mathcal{O}(B^{7})$. 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 $\mathcal{O}(B^{3} \log^{2} B)$ fast SGL Fourier transform and inverse,
and an outlook on future developments.
