this condition is restrictive, and even in very simple settings such as for the case of band-limited functions, it is not satisfied. Nevertheless, in [@dadedelastte17], it has been shown that there is a way out. Instead of using a classical $L_1$-space as the space of generalized test functions, one can work with the weaker concept of Fréchet spaces. Then, more or less all the basic steps to establish the associated coorbit spaces can be performed. We refer to Sect. \[sec:overview\] for brief discussion of this approach.

However, in [@dadedelastte17] one issue remained open, namely the construction of atomic decompositions for the resulting coorbit spaces. This is exactly the problem we are concerned with here. As a surprise, it turns out that this part of the coorbit space theory does not directly carry over to the Fréchet setting. There are two essential differences: First of all, a synthesis map can be constructed, but only at the price that the integrability parameters of the discrete norms on the coefficient spaces and of the coorbit norms are different. At first sight, this might look strange, but in the setting of non-integrable kernels this is in a certain sense not too surprising. Indeed, in 