Deformations and Balian–Low theorems for Gabor frames on the adeles

Abstract

We generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group G. More precisely, we show that Gabor frames over lattices in the time-frequency plane of G with windows in the Feichtinger algebra are stable under small deformations of the lattice by an automorphism of G×Gˆ. The topology we use on the automorphisms is the Braconnier topology. We characterize the groups in which the Balian–Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component. This generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform on locally compact abelian groups. We apply our results to a class of number-theoretic groups, including the adele group associated to a global field.