The Balian–Low theorem for a new kind of Gabor system

Abstract

The Balian–Low theorem expresses the fact that time–frequency concentration is incompatible with non-redundancy for Gabor systems. In this article, the Balian–Low theorem is established for a new kind of Gabor systems associated with a phase function θ(t) satisfying certain assumptions. Meanwhile, some properties of the corresponding generalized Zak transform are shown and explicit examples of the phase function θ(t) are provided.

Keywords:

• Balian–Low theorem
• generalized Zak transform
• Gabor system
• frame
• phase function

AMS Subject Classifications::

• 42C15
• 33C90
• 94A12

Acknowledgements

The first author is the recipient of a postdoctoral grant from FCT (Portugal) with grant No. SFRH/BPD/46250/2008. This research is partially supported by Centro de Investigação e Desenvolvimento em Matemática e Aplicações of the University of Aveiro and in part by the Foundation of Hubei Educational Committee (No. Q20091004) and the NSFC (No. 11026056).