Fast SGL Fourier transforms for scattered data

Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l+1\mathrm{/}2)}(r^2)r^l{Y}_{lm}(\vartheta ,\phi )$, $|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 polynomial basis of the space $L^2$ on ${\mathbb{R}}^3$ with radial Gaussian (multivariate Hermite) weight $\exp (-r^2)$. We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in ${\mathbb{R}}^3$. In this paper, we present fast SGL Fourier transforms for scattered data. The idea is to employ well-known basal fast algorithms to determine a three-dimensional trigonometric polynomial that coincides with the bandlimited function of interest where the latter is to be evaluated. This trigonometric polynomial can then be evaluated efficiently using the well-known non-equispaced FFT (NFFT). We proof an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.