\[sec:DiscretziationUnderAssumptions\] we present satisfactory discretization results with the aid of an additional kernel $W$. Indeed, in the Theorems \[thm:GoodKernelBanachFrameCoorbit\] and \[thm:GoodKernelAtomicDecompositionCoorbit\], respectively, we show that atomic decompositions and Banach frames with uniform bounds can be constructed, just as in the context of the classical coorbit theory.

An Overview {#sec:overview}
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Throughout this paper, $G$ denotes a fixed locally compact second countable group with left *Haar measure* $\beta$ and *modular function* $\Delta$. For a definition of these terms, we refer to [@fol95]. We simply write $\int_G f(x)~dx$ instead of $\int_G f(x)~d\beta(x)$ and we denote by $ L_0(G) $ the space of Borel-measurable functions. Given $f\in L_0(G)$ the functions $\widecheck{f}$ and $\overline{f}$ are $$\widecheck{f}(x)=f(x^{-1}), \qquad \overline{f}(x)=\overline{f(x)},$$ and for all $x\in G$ the left and right regular representations $\lambda$ and $\rho$ act on $f$ as $$\begin{aligned}
{2}
  \lambda(x)f\,(y) & =f(x^{-1}y) \qquad &\text{a.e }y\in G,\\
\rho(x)f\,(y) & =f(yx) \qquad &\text{a.e }y\in G.\end{aligned}$$ Finally, the convolution $f\ast g$ between $f,g\in  L_0(G) $ is the function $$f\ast g(x)
  = \int_G f(y) g(y^{-1} x) ~dy
  = \int_G f(y) \cdot (\lambda(x)\widecheck{g})(y) ~dy
  \qquad \text{a.e. } x \in G,$$ provided that, for almost all $x \in G$, the function $y \mapsto f(y) \cdot (\lambda(x) \widecheck{g})(y)$ is integrable.

Furthermore, given 