52 research outputs found
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
operations. The non-oscillatory component
has a multiresolution representation via a linear combination of Gaussians and
is applied efficiently in space.Comment: 10 pages, 3 figure
Nonlinear inversion of a band-limited Fourier transform
AbstractWe consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach
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