The Gabor frame as a discretization for the 2D transverse-electric scattering-problem domain integral equation

We apply the Gabor frame as a projection method to numerically solve a 2D ransverse-electric-polarized domain-integral equation for a homogeneous medium. Since the Gabor frame is spatially as well as spectrally very well convergent, it is convenient to use for solving a domain integral equation. The mixed spatial and spectral nature of the Gabor frame creates a natural and fast way to Fourier transform a function. In the spectral domain we employ a coordinate scaling to smoothen the branchcut found in the Green function. We have developed algorithms to perform multiplication and convolution efficiently, scaling as O(N log N ) on the number of Gabor coefficients, yielding an overall algorithm that also scales as O(N log N ).