On representations of the Helmholtz Green's function

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 $O(k^d\log k)$ 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 $O(c_1{k}^{-1}+c_2{k}^{-1}{\log }_{10}k)$, for some constants $c_1$, $c_2$, whereas the propagating component can be observed at large distances.