The Balian-Low Theorem is a strong form of the uncertainty principle for Gabor systems that form orthonormal or Riesz bases for L^2(R). In this talk we will discuss the Balian-Low Theorem in the setting of Schauder bases and exact Gabor systems that need not be bases. We will give weak versions of the Balian-Low Theorem for Gabor Schauder bases, and constructively demonstrate that several variants of the BLT can fail for Gabor Schauder bases that are not Riesz bases. We also give new (p,q)-Balian-Low Theorems for exact Gabor systems, and discuss some related open problems.

This is joint work with Alex Powell (Vanderbilt University).