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Journal of Fourier Analysis and Applications

, Volume 23, Issue 6, pp 1311–1347 | Cite as

Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements

  • Ben Adcock
  • Milana Gataric
  • Anders C. Hansen
Article

Abstract

We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first k derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as k increases, it grows linearly in \(k+1\) with the constant of proportionality \(1/\mathrm {e}\). Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at \(k+1\) times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval \(s+1\) bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in \(s+1\) for large s, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large s, we obtain the same result as in the case of univariate sampling with s derivatives.

Keywords

Nonuniform sampling Derivative sampling Bunched sampling Frames Sampling density 

Mathematics Subject Classification

42C15 94A20 41A05 

Notes

Acknowledgments

The authors would like to thank Akram Aldroubi, Karlheinz Gröchenig, Maarten Van De Hoop and Ilya Krishtal for useful discussions. Additionally, the authors would like to thank to the participants of the ICERM Research Cluster “Computational Challenges in Sparse and Redundant Representations” for providing a stimulating and interactive research environment. BA was supported by the NSF DMS Grant 1318894. MG and AH were supported by the EPSRC Grant EP/N014588/1 for the EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging. AH was also supported by a Royal Society University Research Fellowship as well as the EPSRC Grant EP/L003457/1.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Ben Adcock
    • 1
  • Milana Gataric
    • 2
  • Anders C. Hansen
    • 3
    • 4
  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  3. 3.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  4. 4.Department of MathematicsUniversity of OsloOsloNorway

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