ar X iv : 0 70 4 . 14 87 v 1 [ m at h . C A ] 1 1 A pr 2 00 7 Wavelet frames , Bergman spaces and Fourier transforms of Laguerre functions

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 Gröchenig and Lyubarskii in Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007). Building on Seip ́s work 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 GröchenigLyubarskii theorem, the density increases with n, and in the special case of the hyperbolic lattice in the upper half plane it is given by b log a < 4π 2n+ α , where α is the parameter of the Laguerre function.