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.

Published online:
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
     author = {Gr\"ochenig, Karlheinz and Lyubarskii, Yurii},
     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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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.

[1] Akhiezer, N.I. Elements of the Theory of Elliptic Functions, Amer. Math. Soc., Providence, RI, 1990

[2] Chistyakov, G.; Lyubarskii, Yu.; Pastur, L. On completeness of random exponentials in the Bargmann–Fock space, J. Math. Phys., Volume 42 (2001) no. 8, pp. 3754-3768

[3] Daubechies, I. The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, Volume 36 (1990) no. 5, pp. 961-1005

[4] Daubechies, I.; Landau, H.J.; Landau, Z. Gabor time-frequency lattices and the Wexler–Raz identity, J. Fourier Anal. Appl., Volume 1 (1995) no. 4, pp. 437-478

[5] Feichtinger, H.G.; Zimmermann, G. A Banach space of test functions for Gabor analysis, Gabor Analysis and Algorithms, Birkhäuser Boston, Boston, MA, 1998, pp. 123-170

[6] Folland, G.B. Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989 (x+277 pp)

[7] Gabor, D. Theory of communication, J. IEE (London), Volume 93 (1946) no. III, pp. 429-457

[8] Gröchenig, K. Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001 (xvi+359 pp)

[9] Hayman, W.K. The local growth of the power series: a survey of the Wiman–Valiron method, Canad. Math. Bull., Volume 17 (1974) no. 3, pp. 317-358

[10] Janssen, A.J.E.M. Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl., Volume 1 (1995) no. 4, pp. 403-436

[11] Janssen, A.J.E.M.; Strohmer, T. Hyperbolic secants yield Gabor frames, Appl. Comp. Harm. Anal. (2001)

[12] Lyubarski, Yu. Frames in the Bargmann space of entire functions, Entire and Subharmonic Functions, Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 167-180

[13] Seip, K.; Wallstén, R. Density theorems for sampling and interpolation in the Bargmann–Fock space. II, J. Reine Angew. Math., Volume 429 (1992), pp. 107-113

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