Some results on the lattice parameters of quaternionic Gabor frames

TL;DR

This paper extends Gabor frames to the quaternionic setting, introduces a two-sided quaternionic Fourier transform, and establishes density conditions and a counterexample to the Gabor conjecture in this context.

Contribution

It provides the first non-trivial generalization of Gabor frames to quaternions, including new density results and the application of the quaternionic Zak transform.

Findings

Established a Heisenberg uncertainty principle for quaternionic Gabor frames.

Derived necessary and sufficient density conditions for quaternionic Gabor frames.

Proved that the Gabor conjecture does not hold in the quaternionic setting.

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.