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Approximately dual Gabor frames and almost perfect reconstruction based on a class of window functions

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Abstract

It is a well-known problem in Gabor analysis how to construct explicitly given dual frames associated with a given frame. In this paper we will consider a class of window functions for which approximately dual windows can be calculated explicitly. The method makes it possible to get arbitrarily close to perfect reconstruction by allowing the modulation parameter to vary. Explicit estimates for the deviation from perfect reconstruction are provided for some of the standard functions in Gabor analysis, e.g., the Gaussian and the two-sided exponential function.

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Acknowledgments

The authors would like to thank the reviewers for several useful suggestions, which improved the presentation of the results.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800, Lyngby, Denmark

    Ole Christensen

  2. Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, 5600, Eindhoven, The Netherlands

    Augustus J. E. M. Janssen

  3. Department of Mathematical Sciences, UNIST, 50 UNIST-gil, Ulsan, 44919, Republic of Korea

    Hong Oh Kim

  4. Department of Mathematics, Yeungnam University, 280 Daehak-Ro, Gyeongsan, Gyeongbuk, 38541, Republic of Korea

    Rae Young Kim

Authors
  1. Ole Christensen
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  2. Augustus J. E. M. Janssen
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  3. Hong Oh Kim
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  4. Rae Young Kim
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Corresponding author

Correspondence to Rae Young Kim.

Additional information

Communicated by: Gitta Kutyniok

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B02009954).

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Christensen, O., Janssen, A.J.E.M., Kim, H.O. et al. Approximately dual Gabor frames and almost perfect reconstruction based on a class of window functions. Adv Comput Math 44, 1519–1535 (2018). https://doi.org/10.1007/s10444-018-9595-7

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  • DOI: https://doi.org/10.1007/s10444-018-9595-7

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