Analysis of Shearlet Coorbit Spaces in arbitrary Dimensions using Coarse Geometry

In order to analyze anisotropic information of signals, the shearlet transform has been introduced as class of directionally selective wavelet transforms. One way of describing the approximation-theoretic properties of such generalized wavelet systems relies on coorbit spaces, i.e., spaces defined in terms of sparsity properties with respect to the system. In higher dimensions, there are several distinct possibilities for the definition of shearlet systems, and their approximation-theoretic properties are currently not well-understood.In this note, we investigate shearlet systems in higher dimensions derived from two particular classes of shearlet groups, the standard shearlet group and the Toeplitz shearlet group. We want to show that different groups lead to different approximation theories. The analysis of the associated coorbit spaces relies on an alternative description via decomposition spaces that was recently established.For a shearlet group, this identification is based on a covering of the associated dual orbit induced by the shearlet group. The geometry of the sets in this covering is the determining factor for the associated decomposition space. We will see that the orbit can be equipped with a metric structure that encodes essential properties of this covering. The orbit map then allows to compare the geometric properties of coverings induced by different groups without the need to explicitly compute the respective coverings, which gets increasingly difficult for higher dimensions.This argument relies on a rigidity theorem which states that geometrically incompatible coverings lead to different decomposition spaces in almost all cases.