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Comptes Rendus Mathématique
Volume 357, n° 2
pages 188-199 (février 2019)
Doi : 10.1016/j.crma.2018.12.004
Received : 27 July 2018 ;  accepted : 7 January 2019
Time-frequency analysis on the adeles over the rationals
Analyse des temps-fréquences sur l'anneau des adèles des rationnels
 

Ulrik B.R. Enstad a , Mads S. Jakobsen b , Franz Luef b
a University of Oslo, Department of Mathematics, Oslo, Norway 
b Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway 

Abstract

We show that the construction of Gabor frames in   with generators in   and with respect to time-frequency shifts from a rectangular lattice   is equivalent to the construction of certain Gabor frames for   over the adeles over the rationals and the group  . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on   with the construction of certain Heisenberg modules.

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Résumé

Nous montrons que la construction de trames (ou repères) de Gabor de   avec générateurs dans   et des décalages de temps-fréquence dans un réseau rectangulaire   est équivalente à la construction de certaines trames de Gabor pour   sur les adèles des rationnels avec le groupe  . Nous analysons également les relations entre la construction de trames de Gabor sur les adèles et sur   et la construction de certains modules de Heisenberg.

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