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
[Trames d'exponentielles et sous-multipavages dans les groupes abéliens localement compacts]

Présenté par : 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.

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 :

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

@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/

Bibliographie
Cité par

[1] E. Agora; J. Antezana; C. Cabrelli Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups, Adv. Math., Volume 285 (2015), pp. 454-477

[2] D. Barbieri; E. Hernández; A. Mayeli Lattice sub-tilings and frames in LCA groups, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2017) no. 2, pp. 193-199

[3] C. Cabrelli; V. Paternostro Shift-invariant spaces on LCA groups, J. Funct. Anal., Volume 258 (2010) no. 6, pp. 2034-2059

[4] J. Feldman; F.P. Greenleaf Existence of Borel transversals in groups, Pac. J. Math., Volume 25 (1968), pp. 455-461

[5] B. Fuglede Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal., Volume 16 (1974), pp. 101-121

[6] S. Grepstad; N. Lev Multi-tiling and Riesz basis, Adv. Math., Volume 252 (2014) no. 15, pp. 1-6

[7] D. Han; K. Kornelson; D. Larson; E. Weber Frames for Undergraduates, Student Mathematical Library, vol. 40, American Mathematical Society, 2007

[8] E. Hewitt; K.A. Ross Abstract Harmonic Analysis, Vol. I, Structure of Topological Groups, Integration Theory, Group Representations, Springer, 1979

[9] M. Kolountzakis Multiple lattice tiles and Riesz bases of exponentials, Proc. Amer. Math. Soc., Volume 143 (2015), pp. 741-747

[10] S. Pedersen Spectral theory of commuting self-adjoint partial differential operators, J. Funct. Anal., Volume 73 (1987), pp. 122-134

[11] R.M. Young Introduction to Nonharmonic Fourier Series, Academic Press, 1980

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