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

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.

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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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Correspondence to Milana Gataric.

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Communicated by Hans G. Feichtinger.

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Adcock, B., Gataric, M. & Hansen, A.C. Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements. J Fourier Anal Appl 23, 1311–1347 (2017). https://doi.org/10.1007/s00041-016-9504-8

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Keywords

  • Nonuniform sampling
  • Derivative sampling
  • Bunched sampling
  • Frames
  • Sampling density

Mathematics Subject Classification

  • 42C15
  • 94A20
  • 41A05