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

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