Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets AbstractThe 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{\mathbb{R}^{d}}admitting common connected fundamental domains of the type N⁢[0,1)d{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{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{N[0,1)^{2}}, where N is an invertible matrix. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

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

Loading next page...
 
/lp/de-gruyter/gabor-orthonormal-bases-generated-by-indicator-functions-of-h0LdcAqpc6
Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2016-0087
Publisher site
See Article on Publisher Site

Abstract

AbstractThe 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{\mathbb{R}^{d}}admitting common connected fundamental domains of the type N⁢[0,1)d{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{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{N[0,1)^{2}}, where N is an invertible matrix.

Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Apr 1, 2018

References