Discrete Quaternionic (Multi-window) Gabor Systems

TL;DR

This paper investigates quaternionic multi-window Gabor systems in discrete space, focusing on frame conditions, orthonormal bases, duality, and stability, with particular attention to real-valued generating sequences.

Contribution

It provides new characterizations and conditions for quaternionic Gabor systems to form frames, bases, and dual systems, extending classical Gabor analysis into quaternionic settings.

Findings

Characterization of quaternionic Gabor frames

Conditions for orthonormal bases in quaternionic Gabor systems

Analysis of duality and stability in quaternionic Gabor systems

Abstract

The aim of this work is to study (Multi-window) Gabor systems in the space \(\ell^2(\mathbb{Z} \times \mathbb{Z}, \mathbb{H})\), denoted by $\mathcal{G}(g,L,M,N)$, and defined by: \[ \left\{ (k_1,k_2)\in \mathbb{Z}^2\mapsto e^{2\pi i \frac{m_1}{M}k_1} g_l(k - nN) e^{2\pi j \frac{m_2}{M}k_2} \right\}_{l \in \mathbb{N}_L, (m_1, m_2) \in \mathbb{N}_M^2, n \in \mathbb{Z}^2}, \] where, $L,M,N$ are positive integers, $i,j$ are the imaginary units in the quaternion algebra, and \( \{g_l\}_{l \in \mathbb{N}_L} \subset \ell^2(\mathbb{Z} \times \mathbb{Z}, \mathbb{H}) \). Special emphasis is placed on the case where the sequences \(g_l\) are real-valued. The questions addressed in this work include the characterization of quaternionic Gabor systems that form frames, the characterization of those that are orthonormal bases, and the admissibility of such systems. We also explore necessary and/or sufficient conditions for Gabor frames. The issue of duality is also discussed. Furthermore, we study the stability of these systems.