facts strongly indicate that with the decomposition results stated in Sect. \[sec:discretization\] we have almost reached the ceiling. However, there is still a little bit of flexibility which we can use to improve our results. Indeed, in Sect. \[sec:DiscretziationUnderAssumptions\] we prove that if there exists a second kernel $W$ that satisfies additional smoothness assumptions and acts as the identity by left and right convolution on the reproducing kernel of the representation, then [*uniform*]{} bounds for both, the synthesis and the analysis part, can be obtained. Fortunately, in one important practical application given by the Paley-Wiener spaces such a kernel can be found.

This paper is organized as follows. First of all, in Sect. \[sec:overview\], we recall the construction of coorbit spaces based on non-integrable kernels. We keep the explanation as short as possible and refer to [@dadedelastte17] for further details. Then, in Sect. \[sec:discretization\], we provide first discretization results for the associated coorbit spaces; the main result is Theorem \[maintheorem\]. Then, in Sect. \[sec:Obstructions\], we are concerned with ‘negative’ results. Indeed, in Theorem \[thm:DiscretizationObstruction\] we show that stable decompositions can only be obtained if the right convolution by the reproducing kernel is bounded on the underlying $L_p$-spaces. Finally, in Sect. 