Deformations and Balian-Low theorems for Gabor frames on the adeles (Q2099106)

The authors generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group \(G\). In particular, they 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 \(G \times \widehat{G}\) using the Braconnier topology. Also, they generalize a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform on locally compact abelian groups and characterize the groups in which the Balian-Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component and fails for groups with compact identity component. Finally, the authors introduce the higher-dimensional \(S\)-adeles and apply their main results for LCA groups to a class of number-theoretic groups, including the adele group associated with a global field.