Wavelet frames on groups and hypergroups via discretization of Calderon 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 use Calderon-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 R-n. for an affine extension of the Heisenberg group, and on many commutative hypergroups.