Sparse approximation and sampling recovery on function classes with a structural condition

In nonlinear approximation, it is common to consider the classes of functions given by their expansion coefficients with respect to some basis $\Psi = (\psi_\mathbf k)$, i.e.,
$$ f = \sum_{\mathbf k \in \mathbb Z^d} a_\mathbf k \psi_\mathbf k\,, $$
with certain structural properties imposed on the coefficients $(a_\mathbf k)$, and certain conditions on the basis $(\psi_\mathbf k)$. A classical example are absolutely convergent series $\sum_{\mathbf k \in \mathbb Z^d} |a_\mathbf k| < \infty$, with respect to an orthogonal or a Riesz basis $(\psi_\mathbf k)$, or even a redundant set of functions. Here, we study the classes of functions $\mathbf A_\beta^{r,b}(\Psi,\mathcal G)$ with the property
$$ \Big(\sum_{\mathbf k \in G_j\setminus G_{j-1}} |a_\mathbf k|^\beta\Big)^{1/\beta} \le 2^{-rj} j^b\,, \quad j \in\mathbb{N}\,, $$
where the index sets $\mathcal G = (G_j)$ satisfy $G_{j-1} \subset G_j$, $\cup_{j=1}^\infty G_j = \mathbb Z^d$. It was shown recently that universal sampling discretization and nonlinear sparse approximation are useful in the sampling recovery problem for this type of functions, namely when $(G_j)$ are dyadic cubes or dyadic hyperbolic crosses. In this paper, we generalise these particular results to the classes of functions defined by the index sets $(G_j)$ of a rather general structure.