Deformations of Gabor frames on the adeles and other locally compact abelian groups

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 × G. 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.

The density theorem for projective representations via twisted group von Neumann algebras

“…If C γ is not σ-regular then there exists β ∈ Γ that commutes with λ yet σ(γ, β) = σ(β, γ). But then (15) gives Tr(λ(γ)) = σ(γ, β)σ(β, γ) Tr(λ(γ)) which implies that Tr(λ(γ)) = 0. Suppose now that C γ is both σ-regular and finite, say…”

“…( The above theorem applies e.g. to the case where A is the adele group of a global field which was studied in [14,15].…”

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…Alternatively, for a Schwartz function, the associated density inequalities in Theorem 1.1 are strict [6,37,51]. Balian-Low type theorems for (classes of) nilpotent groups have been obtained in [35,52] and show that the inequalities in Theorem 1.1 are strict for integrable vectors. It should be mentioned that (non-localized) orthonormal bases in the orbit of a nilpotent Lie group could still exist by Theorem 1.2, and even for nilpotent Lie groups not admitting a lattice, cf.…”

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…Balian-Low type theorems for (classes of) nilpotent groups have been obtained in [35,51] and show that the inequalities in Theorem 1.1 are strict for integrable vectors. It should be mentioned that (non-localized) orthonormal bases in the orbit of a nilpotent Lie group could still exist by Theorem 1.2, and even for nilpotent Lie groups not admitting a lattice, cf.…”