Time-frequency analysis on the adeles over the rationals

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

doi:10.1016/j.crma.2018.12.004

Harmonic analysis

Time-frequency analysis on the adeles over the rationals

Presented by: the Editorial Board

Ulrik B.R. Enstad ¹; Mads S. Jakobsen ²; Franz Luef ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 188-199.

Abstract (Original language)
Résumé

We show that the construction of Gabor frames in $L^{2} (R)$ with generators in $S_{0} (R)$ and with respect to time-frequency shifts from a rectangular lattice $α Z \times β Z$ is equivalent to the construction of certain Gabor frames for $L^{2}$ over the adeles over the rationals and the group $R \times Q_{p}$ . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $R \times Q_{p}$ with the construction of certain Heisenberg modules.

Nous montrons que la construction de trames (ou repères) de Gabor de $L^{2} (R)$ avec générateurs dans $S_{0} (R)$ et des décalages de temps-fréquence dans un réseau rectangulaire $α Z \times β Z$ est équivalente à la construction de certaines trames de Gabor pour $L^{2}$ sur les adèles des rationnels avec le groupe $R \times Q_{p}$ . Nous analysons également les relations entre la construction de trames de Gabor sur les adèles et sur $R \times Q_{p}$ et la construction de certains modules de Heisenberg.

Received: 2018-07-27
Accepted: 2019-01-07
Published online: 2019-01-28

DOI: 10.1016/j.crma.2018.12.004

Author's affiliations:

Ulrik B.R. Enstad ¹; Mads S. Jakobsen ²; Franz Luef ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

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     author = {Ulrik B.R. Enstad and Mads S. Jakobsen and Franz Luef},
     title = {Time-frequency analysis on the adeles over the rationals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {188--199},
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     volume = {357},
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     year = {2019},
     doi = {10.1016/j.crma.2018.12.004},
     language = {en},
}

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Ulrik B.R. Enstad; Mads S. Jakobsen; Franz Luef. Time-frequency analysis on the adeles over the rationals. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 188-199. doi : 10.1016/j.crma.2018.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.12.004/

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