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Pseudospectral Fourier reconstruction with IPRM

Authors: Karlheinz Gröchenig and Tomasz Hrycak

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

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.

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Karlheinz Gröchenig and Tomasz Hrycak. Pseudospectral Fourier reconstruction with IPRM. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@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/entities/document/10.4230/DagSemProc.08492.6},
  URN =		{urn:nbn:de:0030-drops-18830},
  doi =		{10.4230/DagSemProc.08492.6},
  annote =	{Keywords: IPRM, Fourier series, inverse methods, pseudospectral methods}
}

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Pseudospectral Fourier reconstruction with IPRM

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