Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we us Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in Rn, for an affine extension of the Heisenberg group, and on many commutative hypergroups.