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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 pp 117–125Cite as

Recovering Piecewise Smooth Functions from Nonuniform Fourier Measurements

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 106))

Abstract

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.

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Notes

  1. 1.

    We remark in passing that the case of critical density δ = 1 can also be addressed [6], but one cannot in general expect stable reconstruction for δ > 1. See also [11, 12].

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Acknowledgements

Ben Adcock acknowledges support from the NSF DMS grant 1318894. Milana Gataric acknowledges support from the UK EPSRC grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. Anders C. Hansen acknowledges support from a Royal Society University Research Fellowship as well as the EPSRC grant EP/L003457/1.

Author information

Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6

    Ben Adcock

  2. CCA, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK

    Milana Gataric

  3. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK

    Anders C. Hansen

Authors
  1. Ben Adcock
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  2. Milana Gataric
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  3. Anders C. Hansen
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Corresponding author

Correspondence to Ben Adcock .

Editor information

Editors and Affiliations

  1. School of Computing , University of Utah, Salt Lake City, Utah, USA

    Robert M. Kirby

  2. School of Computing, University of Utah, Salt Lake City, Utah, USA

    Martin Berzins

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Adcock, B., Gataric, M., Hansen, A.C. (2015). Recovering Piecewise Smooth Functions from Nonuniform Fourier Measurements. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_8

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