What is the best way to separate data using compressive sensing?

In the book Compressed Sensing by Kutyniok et al, the author talks about data separation using sparse representation. In summary, if we have a signal vector

$$x = x_1 + x_2$$

Then, it would be possible to have vectors $$x_1$$ and $$x_2$$ as a result. In order to do that, they use two basis $$\Phi_1$$ and $$\Phi_2$$ on which $$x_1$$ has a sparse representation in $$\Phi_1$$ but not in $$\Phi_2$$ and $$x_2$$ has a sparse representation in $$\Phi_2$$ and not in $$\Phi_1$$.

I am working with thin signals (spikes) and thick signals (sinc functions, gaussians, plateaus). There is a paper that intends to do this but uses sparsity in $$x_1$$ (thin signals) and sparsity in Daubechies 8 (D8) space for thick signals. This last point does not make sense to me since spikes and some thicks signals are sparse in D8 space.

Am I getting wrong something? Do you have any experience separating signals using CS? My mainly concern is what bases should I use to reconstruct thick signals.

Best regards.