Abstract
In this paper we extend the Balian–Low theorem, which is a version of the uncertainty principle for Gabor (Weyl–Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian–Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian–Low theorem for differential operators associated with a symplectic basis for the symplectic form on R2d.
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