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

Presented by: 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

Abstract (Original language)
Résumé

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 $b \log a < 4 π \frac{n + 1}{α + 1}$ .

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, $b \log a < 4 π \frac{n + 1}{α + 1}$ .

Received: 2010-01-21
Accepted: 2011-02-14
Published online: 2011-02-24

DOI: 10.1016/j.crma.2011.02.013

Author's affiliations:

Luis Daniel Abreu ¹

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

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     title = {Wavelet frames with {Laguerre} functions},
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     pages = {255--258},
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     publisher = {Elsevier},
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     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

References
Cited by

[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.

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[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

Cited by 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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