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arXiv · 1807.07011

Time-frequency analysis on the adeles over the rationals

Ulrik B. R. Enstad·Mads S. Jakobsen·Franz Luef
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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.

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Ulrik B. R. Enstad, Mads S. Jakobsen, Franz Luef. 2018-07-18. Time-frequency analysis on the adeles over the rationals. https://doi.org/10.1016/j.crma.2018.12.004

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