A class of general multi-window Gabor frames and its (weak) Gabor dual frames

A general multi-window Gabor frame whose window functions enjoy different time-frequency shifts has potential applications in processing practical signals with rich time-frequency information. It is known from existing literature that one should be careful to construct or define such Gabor frames in the Zak transform domain; they may not even be defined in this domain. Motivated by this, the present paper addresses general multi-window Gabor frames in the time domain. We provide necessary and sufficient conditions for these Gabor frames, characterize their weak Gabor dual frames, and, for comparison, use a Zak transform matrix to characterize (weak) Gabor dual frames for such Gabor frames with rational time-frequency product. Also some examples are provided both in the time and the Zak transform domains.