The continuous wavelet transform is frequently described as a mathematical microscope. In higher dimensions, there is an increasingly larger choice of such microscopes available, which significantly differ in the way that the wavelet (the “lense” of the microscope) is scaled/rotated/sheared etc. by elements of the dilation group. Summarizing recent results, this note presents a unified and comprehensive approach that allows to study approximation-theoretic aspects of such wavelet systems arising from a rather general class of dilation groups, including the shearlet dilation groups in dimension 2 and higher, using the language and results of coorbit theory. The key feature which this unified approach is based on is the so-called dual action of the dilation group.
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