Georgia Institute of Technology College of Sciences

The Gabor system of a function is the set of all of its integer translations and modulations. The Balian-Low Theorem states that the Gabor system of a function which is well localized in both time and frequency cannot form an Riesz basis for $L^2(\mathbb{R})$. An important tool in the proof is a characterization of the Riesz basis property in terms of the boundedness of the Zak transform of the function. In this talk, we will discuss results showing that weaker basis-type properties also correspond to boundedness of the Zak transform, but in the sense of Fourier multipliers. We will also discuss using these results to prove generalizations of the Balian-Low theorem for Gabor systems with weaker basis properties, as well as for shift-invariant spaces with multiple generators and in higher dimensions.