Frames of exponentials and sub-multitiles in LCA groups

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

doi:10.1016/j.crma.2017.12.002

Theory of signals/Harmonic analysis

Frames of exponentials and sub-multitiles in LCA groups

Presented by: Yves Meyer

Davide Barbieri ¹; Carlos Cabrelli ²; Eugenio Hernández ¹; Peter Luthy ³; Ursula Molter ²; Carolina Mosquera 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, Volume 356 (2018) no. 1, pp. 107-113.

Abstract
Résumé

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.

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.

Received: 2017-10-12
Accepted: 2017-12-04
Published online: 2017-12-12

DOI: 10.1016/j.crma.2017.12.002

Author's affiliations:

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

@article{CRMATH_2018__356_1_107_0,
     author = {Davide Barbieri and Carlos Cabrelli and Eugenio Hern\'andez and Peter Luthy and Ursula Molter and Carolina Mosquera},
     title = {Frames of exponentials and sub-multitiles in {LCA} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {107--113},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.12.002},
     language = {en},
}

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Davide Barbieri; Carlos Cabrelli; Eugenio Hernández; Peter Luthy; Ursula Molter; Carolina Mosquera. Frames of exponentials and sub-multitiles in LCA groups. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 107-113. doi : 10.1016/j.crma.2017.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.002/

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