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Improved sparse fourier approximation results: faster implementations and stronger guarantees

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Abstract

We study the problem of quickly estimating the best k-term Fourier representation for a given periodic function f: [0, 2π] → ℂ. Solving this problem requires the identification of k of the largest magnitude Fourier series coefficients of f in worst case k 2 · logO(1) N time. Randomized sublinear-time Monte Carlo algorithms, which have a small probability of failing to output accurate answers for each input signal, have been developed for solving this problem (Gilbert et al. 2002, 2005). These methods were implemented as the Ann Arbor Fast Fourier Transform (AAFFT) and empirically evaluated in Iwen et al. (Commun Math Sci 5(4):981–998, 2007). In this paper we present a new implementation, called the Gopher Fast Fourier Transform (GFFT), of more recently developed sparse Fourier transform techniques (Iwen, Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012). Experiments indicate that GFFT is faster than AAFFT. In addition to our empirical evaluation, we also consider the existence of sublinear-time Fourier approximation methods with deterministic approximation guarantees for functions whose sequences of Fourier series coefficents are compressible. In particular, we prove the existence of sublinear-time Las Vegas Fourier Transforms which improve on the recent deterministic Fourier approximation results of Iwen (Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012) for Fourier compressible functions by guaranteeing accurate answers while using an asymptotically near-optimal number of function evaluations.

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Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Ben Segal

  2. Mathematics Department, Duke University, Durham, NC, 27708, USA

    M. A. Iwen

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  1. Ben Segal
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  2. M. A. Iwen
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Correspondence to M. A. Iwen.

Additional information

Research supported in part by ONR N00014-07-1-0625 and NSF DMS-0847388.

Ben Segal died unexpectedly at the age of 23 in January 2012, after graduating from the University of Minnesota the previous May with degrees in both mathematics and economics. He was a talented, curious, and enthusiastic student who loved mathematics for its innate beauty.

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Segal, B., Iwen, M.A. Improved sparse fourier approximation results: faster implementations and stronger guarantees. Numer Algor 63, 239–263 (2013). https://doi.org/10.1007/s11075-012-9621-7

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