# 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}
}
• Published 28 June 2012
• Computer Science
• SIAM J. Numer. Anal.
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
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
• Computer Science
Multiscale Model. Simul.
• 2009
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
• Computer Science
Multiscale Model. Simul.
• 2009
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)
• Computer Science
• 2003
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.
Fast Fourier Transforms for Nonequispaced Data
• Computer Science
SIAM J. Sci. Comput.
• 1993
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
Accelerating the Nonuniform Fast Fourier Transform
• Computer Science
SIAM Rev.
• 2004
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
• Computer Science
SIAM J. Sci. Comput.
• 2006
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.