instance, the wavelet transform is associated with the affine group whereas the Gabor transform stems from the Weyl-Heisenberg group. We refer e.g., to [@fegr88; @fegr89a] for details.

This connection with group theory paves the way to the application of another very important concept, namely coorbit theory. This theory has been developed by Feichtinger and Gröchenig already in the late 1980’s, see [@fegr88; @fegr89a; @fegr89b; @gro91]. In recent years, coorbit theory has experienced a real renaissance. Among other things, the connections to the various shearlet transforms [@DaDeGrLa] and to the concept of decomposition spaces [@voigt; @FuehrVoigtlaenderCoorbitAsDecomposition] have been investigated.

Based on a square integrable group representation, by means of coorbit space theory it is possible to construct canonical smoothness spaces, the coorbit spaces, by collecting all functions for which the associated voice transform has a certain decay. Moreover, by discretizing the underlying representation, it is possible to obtain atomic decompositions for the coorbit spaces. Moreover, also Banach frames can be constructed.

The coorbit space theory is based on certain assumptions. In particular, it is not enough that the representation is square-integrable, it must also be [*integrable*]{}, i.e., the reproducing kernel must be contained in a weighted $L_1$-space on the group. Unfortunately, 