Wavelet frames with Laguerre functions

Luis Daniel Abreu

doi:10.1016/j.crma.2011.02.013

Complex Analysis/Theory of Signals

Wavelet frames with Laguerre functions
[Frames dʼondelettes et fonctions de Laguerre]

Présenté par : the Editorial Board

Luis Daniel Abreu ¹

¹ CMUC, Departamento de Matemática da Universidade de Coimbra, 3001-454 Coimbra, Portugal

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 255-258.

Résumé
Abstract

Soit le fonction $Φ_{n}^{α}$ de la forme $F Φ_{n}^{α} (t) = t^{\frac{1}{2}} l_{n}^{α} (2 t)$ , ou $l_{n}^{α}$ est une fonction de Laguerre et $Γ (a, b) = {(a^{m} b k, a^{m})}_{k, m \in Z}$ est une reseau hyperbolique. Notre resultat principal dit que, si lʼ ensemble dʼondelettes $W (Φ_{n}^{α}, Γ (a, b))$ est un frame pour $H^{2} (C^{+})$ , alors, $b \log a < 4 π \frac{n + 1}{α + 1}$ .

Consider the functions $Φ_{n}^{α}$ defined as $F Φ_{n}^{α} (t) = t^{\frac{1}{2}} l_{n}^{α} (2 t)$ , where $l_{n}^{α}$ is a Laguerre function and $Γ (a, b) = {(a^{m} b k, a^{m})}_{k, m \in Z}$ is a hyperbolic lattice. We prove that, if the wavelet system $W (Φ_{n}^{α}, Γ (a, b))$ is a frame of $H^{2} (C^{+})$ , then $b \log a < 4 π \frac{n + 1}{α + 1}$ .

Reçu le : 2010-01-21
Accepté le : 2011-02-14
Publié le : 2011-02-24

DOI : 10.1016/j.crma.2011.02.013

Affiliations des auteurs :

Luis Daniel Abreu ¹

¹ CMUC, Departamento de Matemática da Universidade de Coimbra, 3001-454 Coimbra, Portugal

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     author = {Luis Daniel Abreu},
     title = {Wavelet frames with {Laguerre} functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {255--258},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.013},
     language = {en},
}

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Luis Daniel Abreu. Wavelet frames with Laguerre functions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 255-258. doi : 10.1016/j.crma.2011.02.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.013/

Bibliographie
Cité par

[1] L.D. Abreu Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions, Appl. Comput. Harmon. Anal., Volume 29 (2010) no. 3, pp. 287-302

[2] L.D. Abreu, Beurling type density theorems for polyanalytic functions in the unit disc, in preparation.

[3] L.D. Abreu, Super-wavelets versus poly-Bergman spaces, in preparation.

[4] G. Ascensi; J. Bruna Model space results for the Gabor and wavelet transforms, IEEE Trans. Inform. Theory, Volume 55 (2009) no. 5, pp. 2250-2259

[5] K. Gröchenig; Y. Lyubarskii Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007), pp. 157-162

[6] K. Gröchenig; Y. Lyubarskii Gabor (super) frames with Hermite functions, Math. Ann., Volume 345 (2009) no. 2, pp. 267-286

[7] C. Heil; G. Kutyniok Density of weighted wavelet frames, J. Geom. Anal., Volume 13 (2003) no. 3, pp. 479-493

[8] K. Seip Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc., Volume 117 (1993) no. 1, pp. 213-220

[9] K. Seip Beurling type density theorems in the unit disc, Invent. Math., Volume 113 (1993), pp. 21-39

[10] W. Sun Density of wavelet frames, Appl. Comput. Harmon. Anal., Volume 22 (2007) no. 2, pp. 264-272

Cité par Sources :

^☆ This research was partially supported by CMUC/FCT and FCT project “Frame Design” PTDC/MAT/114394/2009, POCI 2010 and FSE.

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