Wojtek Czaja

The Balian--Low theorem for the symplectic form on R^(2d)

 Abstract:

We consider a problem of generalizing the Balian--Low theorem, which is a version of the uncertainty principle for Gabor (Weyl--Heisenberg) systems. There are several directions that may be pursued: one may consider different weights, different spaces, various types of Gabor systems, several generating functions, etc.

We investigate a generalization of BLT to functions of several variables. In particular, we first prove the Balian--Low theorem for arbitrary quadratic forms. Then we generalize further and "define" the Balian--Low theorem to be a statement about the norms of general first order lineardifferential operators applied to the generating function of the Gabor ON basis. The differential operators are associated with a symplectic basis for the symplectic form on R^(2d).