We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in $\mathcal{O}(k^d\log k)$ operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.

Since the Helmholtz Green's function can be viewed as a point source, this partitioning can be interpreted as a splitting into propagating and evanescent components. We show that the non-oscillatory component is significant only in the vicinity of the source at distances $\mathcal{O}(c_1{k}^{-1}+c_2{k}^{-1}{\log }_{10}k)$, for some constants $c_1$, $c_2$, whereas the propagating component can be observed at large distances.

我们考虑自由空间亥姆霍兹格林函数并将其分解为振荡和非振荡（奇异）分量的总和。目标是将原点处实部奇异性的影响与波数k控制的振荡行为分开。振荡分量可以选择在原点处具有任意有限数量的连续导数，并且可以应用于傅里叶空间中的函数$\mathcal{氧}（k^d\mathrm{日志}k）$运营。非振荡分量通过高斯线性组合具有多分辨率表示，并在空间中有效应用。

由于亥姆霍兹格林函数可以被视为点源，因此这种划分可以解释为分裂为传播分量和倏逝分量。我们表明，非振荡分量仅在远处的源附近才显着$\mathcal{氧}（C_1{k}^{-1}+C_2{k}^{-1}{\mathrm{日志}}_{10}k）$，对于一些常数$C_1$,$C_2$，而传播分量可以在很远的距离处观察到。