Characterization of rationally sampled quaternionic dual Gabor frames

This paper addresses quaternionic dual Gabor frames under the condition that the products of time‐frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time‐frequency shift parameters not equal to 12$$ \frac{1}{2} $$, we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.