We present a new version of the fast Gauss transform (FGT) for discrete and continuous sources. Classical Hermite expansions are avoided entirely, making use only of the plane-wave representation of the Gaussian kernel and a new hierarchical merging scheme. For continuous source distributions sampled on adaptive tensor-product grids, we exploit the separable structure of the Gaussian kernel to accelerate the computation. For discrete sources, the scheme relies on the nonuniform fast Fourier transform (NUFFT) to construct near field plane wave representations. The scheme has been implemented for either free-space or periodic boundary conditions. In many regimes, the speed is comparable to or better than that of the conventional FFT in work per gridpoint, despite being fully adaptive.

翻译：我们提出了一种适用于离散与连续源的新型快速高斯变换（FGT）方法。该方法完全摒弃了经典埃尔米特展开，仅利用高斯核的平面波表示与新型分层合并方案。针对自适应张量积网格采样的连续源分布，我们通过高斯核的可分离结构加速计算。对于离散源，方案采用非均匀快速傅里叶变换（NUFFT）构建近场平面波表示。该方法已实现自由空间与周期边界条件两种模式。在多种应用场景下，尽管保持完全自适应性，其每个网格点的计算效率仍可与传统FFT相当甚至更优。