Georgia Institute of Technology College of Sciences

The Balian-Low theorem is a classical result in time-frequency analysis that describes a trade off between the basis properties of a Gabor system and the smoothness and decay of the Gabor window. 

In particular a Gabor system with well localized window cannot be a Riesz basis for the space of finite energy signals.

We explore a few generalizations of this fact in the setting of Riesz bases for subspaces of L^2 and we show that the Gabor space being invariant under additional time-frequency shifts is incompatible with two different notions of smoothness and decay for the Gabor window.