Mathematical Analysis
Gabor frames with Hermite functions
Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 157-162.

We investigate Gabor frames based on a linear combination of Hermite functions Hn. We derive sufficient conditions on the lattice ΛR2 such that the Gabor system {e2πiλ2tHn(tλ1):λ=(λ1,λ2)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.

Nous étudions les propriétés de frame de l' ensemble {e2πiλ2tHn(tλ1):λ=(λ1,λ2)Λ}, où Hn est une fonction de Hermite et Λ est un réseau dans R2. 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.

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DOI: 10.1016/j.crma.2006.12.013
Gröchenig, Karlheinz 1; Lyubarskii, Yurii 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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Gröchenig, Karlheinz; Lyubarskii, Yurii. Gabor frames with Hermite functions. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 157-162. doi : 10.1016/j.crma.2006.12.013. http://www.numdam.org/articles/10.1016/j.crma.2006.12.013/

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