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Dynamical Sampling and Systems from Iterative Actions of Operators

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Frames and Other Bases in Abstract and Function Spaces

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this chapter, we review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form {A n g: g ∈ G, n = 0, 1, 2, …}, where A is a bounded linear operator on a separable complex Hilbert space \(\mathscr{H}\) and G is a countable set of vectors in \(\mathscr{H}\). The system of iterations mentioned above was motivated from the so-called dynamical sampling problem. In dynamical sampling, an unknown function f and its future states A n f are coarsely sampled at each time level n, 0 ≤ n < L, where A is an evolution operator that drives the system. The goal is to recover f from these space-time samples.

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Acknowledgements

This work has been partially supported by NSF/DMS grant 1322099. Akram Aldroubi would like to thank Charlotte Avant and Barbara Corley for their attendance to the comfort and entertainment during the preparation of this manuscript.

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Authors and Affiliations

  1. Vanderbilt University, Nashville, TN, 37240, USA

    Akram Aldroubi & Armenak Petrosyan

Authors
  1. Akram Aldroubi
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  2. Armenak Petrosyan
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Correspondence to Akram Aldroubi .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Aldroubi, A., Petrosyan, A. (2017). Dynamical Sampling and Systems from Iterative Actions of Operators. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_2

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