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Structural Results for Quaternionic Gabor Frames

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Abstract

We study quaternionic Gabor frames based on the two-sided quaternionic windowed Fourier transform. Since classical Hilbert space based methods do not work in this case we introduce appropriated versions of translation and modulation operators. We prove Janssen’s and Walnut’s representations, as well as modified versions of the Wexler–Raz biorthogonality and Ron–Shen duality based on the concept of correlation function. We end up with a characterization of tight quaternionic Gabor frames.

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Acknowledgements

The work of the first author was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project UID/MAT/ 0416/2013.

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Authors and Affiliations

  1. CIDMA-Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    Paula Cerejeiras & Stefan Hartmann

  2. Laboratory of Civil Engineering, Tampere University of Technology, Korkeakoulunkatu 10, 33720, Tampere, Finland

    Heikki Orelma

Authors
  1. Paula Cerejeiras
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  2. Stefan Hartmann
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  3. Heikki Orelma
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Correspondence to Paula Cerejeiras.

Additional information

This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.

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Cerejeiras, P., Hartmann, S. & Orelma, H. Structural Results for Quaternionic Gabor Frames. Adv. Appl. Clifford Algebras 28, 86 (2018). https://doi.org/10.1007/s00006-018-0901-8

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  • DOI: https://doi.org/10.1007/s00006-018-0901-8

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