A Stable and Accurate Butterfly Sparse Fourier Transform

@article{Kunis2012ASA,
  title={A Stable and Accurate Butterfly Sparse Fourier Transform},
  author={Stefan Kunis and Ines Melzer},
  journal={SIAM J. Numer. Anal.},
  year={2012},
  volume={50},
  pages={1777-1800}
}
Recently, the butterfly approximation scheme was proposed for computing Fourier transforms with sparse and smooth sampling in the frequency and spatial domains. We present a rigorous error analysis which shows how the local expansion degree depends on the target accuracy and the nonharmonic bandwidth. Moreover, we show that the original scheme becomes numerically unstable if a large local expansion degree is used. This problem is removed by representing all approximations in a Lagrange-type… 
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References

SHOWING 1-10 OF 21 REFERENCES
Sparse Fourier Transform via Butterfly Algorithm
TLDR
A fast algorithm for computing sparse Fourier transforms with spatial and Fourier data supported on curves or surfaces that can approximate the interaction between a frequency region and a spatial region accurately and compactly using a small number of equivalent sources.
A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators
TLDR
This paper introduces a novel algorithm running in O(N^2 log N) time, i.e., with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm.
Fast Computation of Partial Fourier Transforms
TLDR
Two ecient algorithms for computing the partial Fourier transforms in one and two dimensions are introduced by the wave extrapolation procedure in reection seismology to decompose the summation domain of into simpler components in a multiscale way.
A sparse data fast Fourier transform (SDFFT)
TLDR
The parabolic reflector antenna problem is studied as an example to demonstrate its use in the computation of far-field patterns due to arbitrary aperture antennas and antenna arrays.
An algorithm for the rapid evaluation of special function transforms
Fast Fourier Transforms for Nonequispaced Data
A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval $[ - \pi ,\pi ]$. The schemes of this paper…
Fast algorithms for spherical harmonic expansions, II
Fast algorithms for spherical harmonic expansions, III
  • M. Tygert
  • Computer Science
    J. Comput. Phys.
  • 2010
Accelerating the Nonuniform Fast Fourier Transform
TLDR
This paper observes that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights, of particular value in two- and three- dimensional settings.
Fast Algorithms for Spherical Harmonic Expansions
TLDR
An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the two-dimensional sphere of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform); the performance of the algorithm is illustrated via several numerical examples.
...
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