Time-frequency analysis on the adeles over the rationals

Abstract

We show that the construction of Gabor frames in $L^2(\mathbb{R})$ with generators in ${\mathbf{S}}_0(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $\alpha \mathbb{Z}\times \beta \mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^2$ over the adeles over the rationals and the group $\mathbb{R}\times {\mathbb{Q}}_p$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times {\mathbb{Q}}_p$ with the construction of certain Heisenberg modules.

minor revisions, added more references, added a Balian-Low type result in the form of Proposition 4.4

References in corpus (1)

• Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions

Cited by in corpus (3)

• Heisenberg modules as function spaces
• The density theorem for projective representations via twisted group von Neumann algebras
• The Balian-Low theorem for locally compact abelian groups and vector bundles