A Quantitative Balian-Low Theorem

We study functions generating Gabor Riesz bases on the integer lattice. The classical Balian-Low theorem (BLT) restricts the simultaneous time and frequency localization of such functions. We obtain a quantitative estimate on their Zak transform that extends both this result and the more general (p,q) Balian-Low theorem. Moreover, we establish a family of quantitative amalgam-type Balian-Low theorems that contain both the amalgam BLT and the classical BLT as special cases.