Coarse geometric methods for generalized wavelet approximation theory
H. Führ
Abstract: Generalized wavelet systems, as studied in this talk, arise from the translation action on a family of mother wavelets, chosen to ensure perfect reconstruction. This rather general notion comprises Gabor systems (including the multi-window variety), generalized wavelet systems arising from the action of matrix groups such as shearlets, but also curvelets and related constructions. The approximation theoretic properties of these systems are coded in the associated decomposition spaces, defined by imposing suitably weighted Lp-norms on the transform side. Understanding the dependence of the scale of decomposition spaces on properties of the wavelet system is one of the basic problems of the theory, and generally not well understood. The talk translates this question into a problem from the domain of coarse geometry, concerning quasi-equivalence of suitably defined metrics in frequency domain, and presents several applications.