Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l+1/2)}(r^2)r^l{Y}_{lm}(\vartheta ,\phi )$, $|m|\le l<n\in \mathbb{N}$, $L_{n-l-1}^{(l+1/2)}$ 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 prove an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.

球形高斯-拉格瑞（SGL）基函数，即类型的归一化函数${\mathrm{大号}}_{ñ-升-\mathrm{1个}}^{（升+\mathrm{1个}/2）}（{\mathrm{[R}}^2）{\mathrm{[R}}^{升}{ÿ}_{升米}（\vartheta ，\phi ）$， $|米|\le 升<ñ\in \mathbb{ñ}$， ${\mathrm{大号}}_{ñ-升-\mathrm{1个}}^{（升+\mathrm{1个}/2）}$ 是广义的Laguerre多项式， ${ÿ}_{升米}$ 球谐函数，构成空间的正交多项式基础 ${\mathrm{大号}}^2$ 上 ${\mathbb{[R}}^3$ 径向高斯（多元Hermite）权重 $\mathrm{经验值}（-{\mathrm{[R}}^2）$。最近，我们已经针对SGL基函数描述了基于具有特定网格点的精确正交公式的快速傅里叶变换。${\mathbb{[R}}^3$。在本文中，我们提出了用于分散数据的快速SGL傅立叶变换。想法是采用众所周知的基础快速算法来确定与要评估的带限函数相符的三维三角多项式。然后可以使用众所周知的非等距FFT（NFFT）有效地评估该三角多项式。我们证明了我们算法的误差估计，并在广泛的数值实验中验证了它们的实际适用性。