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Pseudospectral Fourier reconstruction with IPRM

Authors Karlheinz Gröchenig, Tomasz Hrycak

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DagSemProc.08492.6.pdf
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Keywords
  • IPRM
  • Fourier series
  • inverse methods
  • pseudospectral methods

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/DagSemProc.08492.6

Author Details

Karlheinz Gröchenig
Tomasz Hrycak
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