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URN: urn:nbn:de:0030-drops-18830
URL: http://drops.dagstuhl.de/opus/volltexte/2009/1883/

Gröchenig, Karlheinz ; Hrycak, Tomasz

Pseudospectral Fourier reconstruction with IPRM

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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{grchenig_et_al:DSP:2009:1883,
  author =	{Karlheinz Gr{\"o}chenig and Tomasz Hrycak},
  title =	{Pseudospectral Fourier reconstruction with IPRM},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  year =	{2009},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  number =	{08492},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/1883},
  annote =	{Keywords: IPRM, Fourier series, inverse methods, pseudospectral methods}
}

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


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