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Sobolev Duals in Frame Theory and Sigma-Delta Quantization

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An Erratum to this article was published on 19 February 2010

Abstract

A new class of alternative dual frames is introduced in the setting of finite frames for ℝd. These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order \(\mathcal{O}(N^{-r})\) for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Wilmington, Wilmington, NC, 28403, USA

    James Blum & Mark Lammers

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Alexander M. Powell

  3. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Özgür Yılmaz

Authors
  1. James Blum
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  2. Mark Lammers
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  3. Alexander M. Powell
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  4. Özgür Yılmaz
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Corresponding author

Correspondence to Özgür Yılmaz.

Additional information

Communicated by Peter Casazza.

M. Lammers was supported in part by a Cahill Grant from UNCW. A.M. Powell was supported in part by NSF Grant DMS-0811086. Ö. Yılmaz was supported in part by a Discovery Grant from NSERC Canada.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00041-010-9120-y

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Blum, J., Lammers, M., Powell, A.M. et al. Sobolev Duals in Frame Theory and Sigma-Delta Quantization. J Fourier Anal Appl 16, 365–381 (2010). https://doi.org/10.1007/s00041-009-9105-x

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  • DOI: https://doi.org/10.1007/s00041-009-9105-x

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