A Balian-Low type theorem for Gabor Riesz sequences of arbitrary density (Q2682941)

It was shown by \textit{H. G. Feichtinger} and \textit{K. Gröchenig} [J. Funct. Anal. 146, No. 2, 464--495 (1997; Zbl 0887.46017)] that if \(g\in S_{o}(\mathbb{R}^{d})\) (Feichtinger algebra) generates a Gabor frame for \(L^{2}(\mathbb{R}^{d})\) over a lattice of rational density, then the canonical dual window also belongs to \( S_{o}(\mathbb{R}^{d})\). In this paper, the results studied by \textit{C. Cabrelli} et al. [Appl. Comput. Harmon. Anal. 41, No. 3, 677--691 (2016; Zbl 1360.46021)] are extended from lattices of rational density to arbitrary lattices. It is observed that the Zak transform is a powerful tool for analyzing Gabor systems generated by lattices with rational density, yet it is not of much use in the case of irrational density lattices. Instead of applying the Zak transform and thus dealing with functions on \(\mathbb{R}^{2}\), the rich theory of time-frequency analysis is exploited by using the given objects. Several new interesting statements related to time-frequency shift invariance are obtained.