Weighted Beurling Density and Shift-Invariant Gabor Systems

This chapter is concerned with the study of shift-invariant Gabor systems using the notion of weighted Beurling density. We introduce this new notion of Beurling density for collections of weighted sequences in ℝ^d and prove a useful reinterpretation for it. Then we derive a fundamental relationship for weighted Gabor frames with finitely many generators between the weighted Beurling density of the sequences of time-frequency indices, the frame bounds, and the norms of the generators. Finally, we study shift-invariant weighted Gabor systems and prove necessary density conditions for the sequences of time-frequency indices of a Gabor system and its shift-invariant counterpart.