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A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

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Abstract

We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the well-known consistent reconstruction technique (Eldar et al.). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable.

Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.

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Acknowledgements

The authors would like to thank Emmanuel Candès and Hans G. Feichtinger for valuable discussions and input.

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Authors and Affiliations

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

    Ben Adcock

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK

    Anders C. Hansen

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  1. Ben Adcock
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  2. Anders C. Hansen
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Correspondence to Anders C. Hansen.

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Communicated by Thomas Strohmer.

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Adcock, B., Hansen, A.C. A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases. J Fourier Anal Appl 18, 685–716 (2012). https://doi.org/10.1007/s00041-012-9221-x

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  • DOI: https://doi.org/10.1007/s00041-012-9221-x

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