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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Complete algebraic reconstruction of piecewise-smooth functions from Fourier data
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by Dmitry Batenkov PDF
Math. Comp. 84 (2015), 2329-2350 Request permission

Abstract:

In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymptotic rate of convergence—including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K. Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff’s equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a “decimated” set which is evenly spread throughout the spectrum.
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Additional Information
  • Dmitry Batenkov
  • Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
  • Address at time of publication: Department of Computer Science, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel
  • MR Author ID: 881951
  • ORCID: setImmediate$0.09410305223452953$7
  • Email: batenkov@cs.technion.ac.il
  • Received by editor(s): December 2, 2012
  • Received by editor(s) in revised form: November 30, 2013
  • Published electronically: February 19, 2015
  • Additional Notes: This research has been supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2329-2350
  • MSC (2010): Primary 65T40; Secondary 65D15
  • DOI: https://doi.org/10.1090/S0025-5718-2015-02948-2
  • MathSciNet review: 3356028