Super-wavelets versus poly-Bergman spaces

TL;DR

This paper explores the relationship between super-wavelets, poly-Bergman spaces, and sampling sequences, establishing new conditions for wavelet frame existence and connecting wavelet analysis with complex function spaces.

Contribution

It constructs an isometric isomorphism between $L^{2}( ^{+}, ^{n})$ and poly-Bergman spaces, linking wavelet frames to sampling sequences in these spaces.

Findings

Proves a necessary condition $b \\ln a < 2\\pi (n+1)$ for wavelet frame existence.

Establishes a novel connection between wavelet analysis and poly-Bergman spaces.

First known result relating wavelet frames to sampling in poly-Bergman spaces outside analytic functions.

Abstract

Motivated by potential applications in multiplexing and by recent results on Gabor analysis with Hermite windows due to Gr\"{o}chenig and Lyubarskii, we investigate vector-valued wavelet transforms and vector-valued wavelet frames, which constitute special cases of super-wavelets, with a particular attention to the case when the analyzing wavelet vector is related to Fourier transforms of Laguerre functions. We construct an isometric isomorphism between $L^{2}(\mathbb{R}^{+},\mathbf{C}^{n})$ and poly-Bergman spaces, with a view to relate the sampling sequences in the poly-Bergman spaces to the wavelet frames and super-frames with the windows $\Phi_{n}$. One of the applications of the theory is a proof that $b\ln a<2\pi (n+1)$ is a necessary condition for the (scalar) wavelet frame associated to the $\Phi_{n}$ to exist. This seems to be the first known result of this type outside the setting of analytic functions (the case $n=0$, which has been completely studied by Seip in 1993).