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Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

Editorial Board: Aldroubi, Akram / Anker, Jean-Philippe / Aouadi, Saloua / Bahouri, Hajer / Baklouti, Ali / Bakry, Dominique / Baraket, Sami / Ben Abdelghani, Leila / Begehr, Heinrich / Beznea, Lucian / Bezzarga, Mounir / Bonami, Aline / Demailly, Jean-Pierre / Fleckinger, Jacqueline / Gallardo, Leonard / Ismail, Mourad / Jarboui, Noomen / Jouini, Elyes / Karoui, Abderrazek / Kamoun, Lotfi / Kobayashi, Toshiyuki / Maday, Yvon / Marzougui, Habib / Mili, Maher / Mustapha, Sami / Ovsienko, Valentin / Peigné, Marc / Pouzet, Maurice / Radulescu, Vicentiu / Schwartz, Lionel / Sifi, Mohamed / Zaag, Hatem / Zarati, Said

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1869-6090
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Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets

Heidi Burgiel / Vignon Oussa
  • Corresponding author
  • Department of Mathematics & Computer Science, Bridgewater State University, Bridgewater, MA 02325, USA
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Published Online: 2017-08-30 | DOI: https://doi.org/10.1515/apam-2016-0087

Abstract

The main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in d admitting common connected fundamental domains of the type N[0,1)d, where N is an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type N[0,1)d. We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type N[0,1)2, where N is an invertible matrix.

Keywords: Lattices; fundamental domains; tiling; packing; orthonormal bases

MSC 2010: 52C22; 52C17; 42B99; 42C30

References

  • [1]

    K. Grochenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. Google Scholar

  • [2]

    D. Han and Y. Wang, Lattice tiling and the Weyl–Heisenberg frames, Geom. Funct. Anal. 11 (2001), no. 4, 742–758. CrossrefGoogle Scholar

  • [3]

    C. Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2007), 113–166. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    S. Stein and S. Szabo, Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Math. Monogr. 25, Mathematical Association of America, Washington, 1994. Google Scholar

  • [5]

    G. Pfander, P. Rashkov and Y. Wang, A geometric construction of tight multivariate Gabor frames with compactly supported smooth windows, J. Fourier Anal. Appl. 18 (2012), no. 2, 223–239. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2016-09-02

Revised: 2017-08-01

Accepted: 2017-08-08

Published Online: 2017-08-30

Published in Print: 2018-04-01


Citation Information: Advances in Pure and Applied Mathematics, Volume 9, Issue 2, Pages 93–107, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam-2016-0087.

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