2007/04/11 by Luís Daniel Abreu, Luis Daniel Abreu, Abreu, Luis Daniel · 1 citation
Computer Science · Engineering · Mathematics · #30H05 #33C45 #42C15 #42C40 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #math.CA #math.CV #msc:30H05 #msc:33C45 #msc:42C15 #msc:42C40
paper · pdf · doi:10.48550/arxiv.0704.1487
15 pages
arxiv created 2007/04/11 · openalex publication_date 2007/04/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K. Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames by translations and dilations of the Fourier transform of the nth Laguerre function. As in Groechenig-Lyubarskii theorem, the density increases with n, and in the special case of the hyperbolic lattice in the upper half plane it is given by blog a<(4π)/(2n+α), where alpha is the parameter of the Laguerre function.