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Dynamical sampling on finite index sets

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Abstract

We consider bounded operators A acting iteratively on a finite set of vectors {fi: i ∈ I} in a Hilbert space ℌ and address the problem of providing necessary and sufficient conditions for the collection of iterates {Anfi: i ∈ I, n = 0, 1, 2,...} to form a frame for the space ℌ. For normal operators A we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on A to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator A. For example, A needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case.

These results provide a theoretical solution to the so-called dynamical sampling problem where a signal f that is evolving in time through iterates of an operator A is spatially sub-sampled at various times and one seeks to reconstruct the signal f from these spatial-temporal samples.

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Acknowledgements

The authors would like to thank D. Suárez for sharing his knowledge on sequences in the unit disk and the anonymous referee for her/his comments which helped us to improve the manuscript.

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Authors and Affiliations

  1. Departamento De Matemática Facultad De Ciencias Exactas Y Naturales, Universidad De Buenos Aires, Buenos Aires, Argentina

    Carlos Cabrelli, Ursula Molter & Victoria Paternostro

  2. Instituto De Investigaciones Matemáticas Luis A. Santalo (Imas), Conicet-Universidad De Buenos Aires, Buenos Aires, Argentina

    Carlos Cabrelli, Ursula Molter & Victoria Paternostro

  3. Mathematisch-Geographische Fakultät, KU Eichstätt-Ingolstadt, Ostenstrasse 26, Kollegiengebäude I Bau B, 85072, Eichstätt, Germany

    Friedrich Philipp

Authors
  1. Carlos Cabrelli
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  2. Ursula Molter
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  3. Victoria Paternostro
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  4. Friedrich Philipp
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Corresponding author

Correspondence to Victoria Paternostro.

Additional information

The research of the authors was partially supported by UBACyT under grants 20020130100403BA, 20020130100422BA, and 20020150200110BA, by CONICET (PIP 11220110101018), and MinCyT Argentina under grant PICT-2014-1480.

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Cabrelli, C., Molter, U., Paternostro, V. et al. Dynamical sampling on finite index sets. JAMA 140, 637–667 (2020). https://doi.org/10.1007/s11854-020-0099-2

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  • DOI: https://doi.org/10.1007/s11854-020-0099-2

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