Complete algebraic reconstruction of piecewise-smooth functions from fourier data

Dmitry Batenkov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

In this paper we provide a reconstruction algorithm for piecewisesmooth 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.

Original languageEnglish
Pages (from-to)2329-2350
Number of pages22
JournalMathematics of Computation
Volume84
Issue number295
DOIs
StatePublished - 2015
Externally publishedYes

Keywords

  • Eckhoff's conjecture
  • Eckhoff's method
  • Fourier inversion
  • Gibbs phenomenon
  • Nonlinear approximation
  • Piecewise-smooth functions

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