Time-frequency analysis on the adeles over the rationals

TL;DR

This paper establishes a connection between Gabor frames in real analysis and those over the adeles over the rationals, linking time-frequency analysis with number theory and harmonic analysis on adelic groups.

Contribution

It demonstrates the equivalence of Gabor frame constructions in real and adelic settings and explores their relation to Heisenberg modules, advancing the understanding of time-frequency analysis in number-theoretic contexts.

Findings

Gabor frames in $L^{2}( eal)$ relate to frames over adeles and $ eal imes ats_p$

Construction of Gabor frames on adeles connects to Heisenberg modules

Provides a framework linking time-frequency analysis with number theory

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.