The Balian–Low theorem for symplectic lattices in higher dimensions

AbstractThe Balian–Low theorem expresses the fact that time–frequency concentration is incompatible with non-redundancy for Gabor systems that form orthonormal or Riesz bases for L2(R). We extend the Balian–Low theorem for Riesz bases to higher dimensions, obtaining a weak form valid for all sets of time–frequency shifts which form a lattice in R2d, and a strong form valid for symplectic lattices in R2d. For the orthonormal basis case, we obtain a strong form valid for general non-lattice sets which are symmetric with respect to the origin