Quaternionic Gabor frame characterization and the density theorem

The study of quaternionic Gabor systems has interested some mathematicians in recent years. From the literature, we found that most existing results on quaternionic Gabor frames focus on the case of the product of time-frequency shift parameters being equal to 1, and have a gap that the involved quaternionic Gabor systems are all incomplete according to the symmetric real scalar inner product. In this paper, we introduce quaternionic Zak transformation and a class of quaternionic Gabor systems. Under the condition that the products of time-frequency shift parameters are rational numbers, we characterize completeness and frame property of quaternionic Gabor systems in terms of Zak transformation matrices. From this, we derive the density theorem for quaternionic Gabor systems.