Discretization of continuous wavelet transforms and wavelet frames

Wavelets are the main topic of our thesis. Different problems are considered in different chapters.; In Chapter 1 we consider the following. Suppose we have a continuous wavelet transform on some space, for example $Rn$, or a group (e.g. the Heisenberg group $Hn$), or a hypergroup (e.g. the Bessel-Kingman hypergroup of order α). This continuous wavelet transforms arise from square-integrable representations in the case of groups, and from the properties of convolutions and Fourier transforms, in the original spirit of Calderón, at least in the Euclidean setting, but also in the context of some groups and hypergroups. The point here is that these continuous transforms exists in many contexts, can be derived in various ways, but all lead to very similar continuous reproducing formulas, that express a function as a continuous superposition of dilates and translates of a single wavelet function. Given these continuous systems of the wavelets, we consider the problem of sampling dilations and translations in such a way that the “extracted” countable wavelet subsystem is still complete and is a frame. This is related to many different problems, such as regular and irregular sampling, reproducing kernel Hilbert spaces (in complex analysis, or arising from coherent states representations), existence of discrete wavelets on groups and hypergroups, stability of wavelet systems under perturbations. By using techniques based on Calderón-Zygmund operators on spaces of homogeneous type, we prove that discretization of continuous wavelet is possible in very general situations, and we obtain strong convergence results for the associated discrete expansions.; In Chapter 2 we present the joint work with C. K. Chui, W. Czaja and G. Weiss, about the characterization of discrete wavelet frames in $Rn$, with arbitrary matrix dilations and translations. We also consider the so-called “oversampling problem” and present a general solution.; In Chapter 3 we present our paper on a new family of biorthogonal wavelets with dilation factor M, and on the study of their regularity. The properties of these wavelets seem promising in view of their applications to digital signal processing and compression.