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Title: Some results on the lattice parameters of quaternionic Gabor frames
Author: Hartmann, Stefan
Keywords: Quaternionic Gabor system
Quaternionic Gabor frames
Windowed quaternionic Fourier transform
Quaternionic Zak transform
Critical density
Issue Date: 2015
Publisher: Springer
Abstract: Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case.
Peer review: yes
DOI: 10.1007/s00006-015-0587-0
ISSN: 0188-7009
Appears in Collections:CIDMA - Artigos
CHAG - Artigos

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