Zum Hauptinhalt springen
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • und EMAIL logo
Veröffentlicht/Copyright: 30. August 2017
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Advances in Pure and Applied Mathematics
Aus der Zeitschrift Advances in Pure and Applied Mathematics Band 9 Heft 2

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 ℝd 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

Acknowledgements

The authors wish to thank their former student Ashley Erwin whose Honors Thesis led to the completion of this research project.

References

[1] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0003-1Suche in Google Scholar

[2] D. Han and Y. Wang, Lattice tiling and the Weyl–Heisenberg frames, Geom. Funct. Anal. 11 (2001), no. 4, 742–758. 10.1007/PL00001683Suche in Google Scholar

[3] C. Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2007), 113–166. 10.1007/s00041-006-6073-2Suche in Google 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. 10.5948/UPO9781614440246Suche in 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. 10.1007/s00041-011-9198-xSuche in Google Scholar

Received: 2016-9-2
Revised: 2017-8-1
Accepted: 2017-8-8
Published Online: 2017-8-30
Published in Print: 2018-4-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Transitivities of maps on G-spaces
  3. Lp-L2 estimates for solutions of the wave equation associated to the Grushin operator
  4. Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets
  5. Riesz potentials of Radon measures associated to reflection groups
  6. Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure
  7. An efficient method to compute the Moore–Penrose inverse
  8. The Thirring model in spaces of analytic functions
https://doi.org/10.1515/apam-2016-0087
Schlagwörter für diesen Artikel
Lattices; fundamental domains; tiling; packing; orthonormal bases

Artikel in diesem Heft

  1. Frontmatter
  2. Transitivities of maps on G-spaces
  3. Lp-L2 estimates for solutions of the wave equation associated to the Grushin operator
  4. Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets
  5. Riesz potentials of Radon measures associated to reflection groups
  6. Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure
  7. An efficient method to compute the Moore–Penrose inverse
  8. The Thirring model in spaces of analytic functions
Heruntergeladen am 22.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/apam-2016-0087/html
Button zum nach oben scrollen