The Balian–Low theorem for symplectic lattices in higher dimensions

Abstract The 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 L 2 ( 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 R 2d , and a strong form valid for symplectic lattices in R 2d . For the orthonormal basis case, we obtain a strong form valid for general non-lattice sets which are symmetric with respect to the origin.

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