Frames of exponentials and sub-multitiles in LCA groups

Barbieri, Davide; Cabrelli, Carlos; Hernández, Eugenio; Luthy, Peter; Molter, Ursula; Mosquera, Carolina

doi:10.1016/j.crma.2017.12.002

Theory of signals/Harmonic analysis

Frames of exponentials and sub-multitiles in LCA groups
[Trames d'exponentielles et sous-multipavages dans les groupes abéliens localement compacts]

Présenté par : Yves Meyer

Barbieri, Davide ¹ ; Cabrelli, Carlos ² ; Hernández, Eugenio ¹ ; Luthy, Peter ³ ; Molter, Ursula ² ; Mosquera, Carolina ²

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain
² Departamento de Matemática, FCEyN, Universidad de Buenos Aires and IMAS-UBA-CONICET, Argentina
³ College of Mount Saint Vincent, Bronx, NY, USA

Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 107-113.

Résumé
Abstract

Dans cette note, nous étudions l'existence de trames d'exponentielles pour $L^{2} (Ω)$ dans le cadre des groupes abéliens localement compacts. Notre résultat principal montre que les propriétés de sous-multipavage de $Ω \subset \hat{G}$ par rapport à un réseau Γ de $\hat{G}$ garantissent l'existence d'une trame d'exponentielles dont les fréquences appartiennent à une union finie de translatés de l'annulateur de Γ. On prouve aussi la réciproque de ce résultat et on donne des conditions pour l'existence de ces trames. Ces conditions étendent des résultats récents sur les bases de Riesz d'exponentielles et les multipavages au cadre des trames.

In this note, we investigate the existence of frames of exponentials for $L^{2} (Ω)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $Ω \subset \hat{G}$ with respect to a uniform lattice Γ of $\hat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

Reçu le : 2017-10-12
Accepté le : 2017-12-04
Publié le : 2017-12-12

DOI : 10.1016/j.crma.2017.12.002

Affiliations des auteurs :

Barbieri, Davide ¹ ; Cabrelli, Carlos ² ; Hernández, Eugenio ¹ ; Luthy, Peter ³ ; Molter, Ursula ² ; Mosquera, Carolina ²

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     author = {Barbieri, Davide and Cabrelli, Carlos and Hern\'andez, Eugenio and Luthy, Peter and Molter, Ursula and Mosquera, Carolina},
     title = {Frames of exponentials and sub-multitiles in {LCA} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {107--113},
     publisher = {Elsevier},
     volume = {356},
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     year = {2018},
     doi = {10.1016/j.crma.2017.12.002},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2017.12.002/}
}

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Barbieri, Davide; Cabrelli, Carlos; Hernández, Eugenio; Luthy, Peter; Molter, Ursula; Mosquera, Carolina. Frames of exponentials and sub-multitiles in LCA groups. Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 107-113. doi : 10.1016/j.crma.2017.12.002. http://archive.numdam.org/articles/10.1016/j.crma.2017.12.002/

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