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Principal shift-invariant spaces with extra invariance nearest to observed data

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Abstract

Given a set of functions \({\mathcal{F}=\{f_1, \dots, f_m\}\subset L^2(\mathbb{R}),}\) we construct a principal shift-invariant space V nearest to \({\mathcal{F}}\) in the sense that V minimizes the expression

$$\sum_{i=1}^{m}\|f_i-P_{V}f_i\|^2,$$

among all the principal shift-invariant spaces with an orthonormal generator which are also translation invariant, or among all the principal shift-invariant spaces with an orthonormal generator which are also \({\frac{1}{n}\mathbb{Z}}\) -invariant for some fixed \({n\in\mathbb{N}}\) .

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Akram Aldroubi & Haichao Wang

  2. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115, USA

    Ilya Krishtal

  3. Department of Mathematics, Ecole Normale Superieure de Lyon, Lyon, France

    Romain Tessera

Authors
  1. Akram Aldroubi
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  2. Ilya Krishtal
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  3. Romain Tessera
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  4. Haichao Wang
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Corresponding author

Correspondence to Ilya Krishtal.

Additional information

Dedicated to Hans Georg Feichtinger on the occasion of his 60th Birthday.

The research of A. Aldroubi and I. Krishtal is supported in part by NSF Grants DMS-0807464 and DMS-0908239, respectively.

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Aldroubi, A., Krishtal, I., Tessera, R. et al. Principal shift-invariant spaces with extra invariance nearest to observed data. Collect. Math. 63, 393–401 (2012). https://doi.org/10.1007/s13348-011-0047-7

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  • DOI: https://doi.org/10.1007/s13348-011-0047-7

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