Gabor frames with Hermite functions

Karlheinz Gröchenig; Yurii Lyubarskii

doi:10.1016/j.crma.2006.12.013

Mathematical Analysis

Gabor frames with Hermite functions
[Frames de Weyl–Heisenberg et fonctions d'Hermite]

Présenté par : Yves Meyer

Karlheinz Gröchenig ¹ ; Yurii Lyubarskii ²

¹ Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
² Department of Mathematics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 157-162.

Résumé
Abstract

Nous étudions les propriétés de frame de l' ensemble ${e^{2 π i λ_{2} t} H_{n} (t - λ_{1}) : λ = (λ_{1}, λ_{2}) \in Λ}$ , où $H_{n}$ est une fonction de Hermite et Λ est un réseau dans $R^{2}$ . Nous donnons des conditions suffisantes sur la densité de Λ pour que la propriété de frame soit satisfaite. Un contre-exemple suggère que nos conditions sont aussi nécessaires. Les outils principaux sont des estimations de croissance pour la fonction σ de Weierstrass et un nouveau type d'interpolation dans l'espace de Bargmann–Fock.

We investigate Gabor frames based on a linear combination of Hermite functions $H_{n}$ . We derive sufficient conditions on the lattice $Λ \subseteq R^{2}$ such that the Gabor system ${e^{2 π i λ_{2} t} H_{n} (t - λ_{1}) : λ = (λ_{1}, λ_{2}) \in Λ}$ is a frame. An example supports our conjecture that our conditions are sharp. The main tools are growth estimates for the Weierstrass σ-function and a new type of interpolation problem for entire functions on the Bargmann–Fock space.

Reçu le : 2006-05-04
Accepté le : 2006-12-12
Publié le : 2007-01-17

DOI : 10.1016/j.crma.2006.12.013

Affiliations des auteurs :

Karlheinz Gröchenig ¹ ; Yurii Lyubarskii ²

¹ Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
² Department of Mathematics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

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     author = {Karlheinz Gr\"ochenig and Yurii Lyubarskii},
     title = {Gabor frames with {Hermite} functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--162},
     publisher = {Elsevier},
     volume = {344},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crma.2006.12.013},
     language = {en},
}

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Karlheinz Gröchenig; Yurii Lyubarskii. Gabor frames with Hermite functions. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 157-162. doi : 10.1016/j.crma.2006.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.12.013/

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