Bounded Fourier multipliers with applications to Balian-Low type theorems

Analysis Seminar
Wednesday, September 27, 2017 - 13:55
1 hour (actually 50 minutes)
Skiles 005
Georgia Tech
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.