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Mathematics of Computation

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Complete algebraic reconstruction of piecewise-smooth functions from Fourier data

Author: Dmitry Batenkov
Journal: Math. Comp. 84 (2015), 2329-2350
MSC (2010): Primary 65T40; Secondary 65D15
Published electronically: February 19, 2015
MathSciNet review: 3356028
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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

Keywords: Fourier inversion, nonlinear approximation, piecewise-smooth functions, Eckhoff’s conjecture, Eckhoff’s method, Gibbs phenomenon
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.
Article copyright: © Copyright 2015 American Mathematical Society