Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions

TL;DR

This paper explores the use of Fourier transforms of Laguerre functions as wavelets in analysis, establishing density conditions for frame construction in Bergman spaces, extending prior work on Hermite functions and Gabor analysis.

Contribution

It introduces a new framework for wavelet analysis using Laguerre functions and derives density conditions for frames in Bergman spaces, expanding the theoretical understanding of sampling sequences.

Findings

Density condition b log a < 4π / (2n + α) for frame construction

Density increases with the order n of Laguerre functions

Extension of Gabor analysis techniques to Bergman spaces

Abstract

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 b\log a<\frac{4\pi}{2n+\alpha}, where alpha is the parameter of the Laguerre function.