The Balian–Low theorem for locally compact abelian groups and vector bundles

Abstract

Let Λ be a lattice in a second countable, locally compact abelian group G with annihilator Λ⊥⊆Gˆ. We investigate the validity of the following statement: For every η in the Feichtinger algebra S0(G), the Gabor system {MτTλη}λ∈Λ,τ∈Λ⊥ is not a frame for L2(G). When Λ is a lattice in G=R, this statement is a variant of the Balian–Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to (G,Λ) is equivalent to the nontriviality of a certain vector bundle over the compact space (G/Λ)×(Gˆ/Λ⊥). We prove this equivalence using Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert C⁎-modules. As an application, we prove a Balian–Low theorem in the new context of the group R×Qp, where Qp denotes the p-adic numbers.