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

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