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Advances in Pure and Applied Mathematics

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

Heidi Burgiel / Vignon Oussa
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  • 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


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


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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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