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License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.08492.6
URN: urn:nbn:de:0030-drops-18830
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2009/1883/
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Gröchenig, Karlheinz ; Hrycak, Tomasz

Pseudospectral Fourier reconstruction with IPRM

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08492.GroechenigKarlheinz.ExtAbstract.1883.pdf (0.1 MB)

Abstract

We generalize the Inverse Polynomial Reconstruction Method (IPRM) for
mitigation of the Gibbs phenomenon by reconstructing a function as an
algebraic polynomial of degree $n-1$ from the function's $m$ lowest
Fourier coefficients ($m ge n$). We compute approximate Legendre
coefficients of the function by solving a linear least squares
problem, and we show that the condition number of the problem does not
exceed $sqrtfrac{m}{{m-alpha_0 n^2}}$, where $alpha_0 =
frac{4sqrt{2}}{pi^2} = 0.573 ldots$. Consequently, whenever
mbox{$m ge n^2$,} the convergence rate of the modified IPRM for an
analytic function is root exponential on the whole interval of
definition. Stability and accuracy of the proposed algorithm are
validated with numerical experiments.

BibTeX - Entry

@InProceedings{grochenig_et_al:DagSemProc.08492.6,
  author =	{Gr\"{o}chenig, Karlheinz and Hrycak, Tomasz},
  title =	{{Pseudospectral Fourier reconstruction with IPRM}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2009/1883},
  URN =		{urn:nbn:de:0030-drops-18830},
  doi =		{10.4230/DagSemProc.08492.6},
  annote =	{Keywords: IPRM, Fourier series, inverse methods, pseudospectral methods}
}

Keywords: IPRM, Fourier series, inverse methods, pseudospectral methods
Collection: 08492 - Structured Decompositions and Efficient Algorithms
Issue Date: 2009
Date of publication: 24.02.2009

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