Sparse Approximation and Sampling Recovery on Function Classes with a Structural Condition

In nonlinear approximation, it is common to consider 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 conditions imposed on the coefficients $$(a_{\mathbf k})$$ and certain conditions imposed on the basis $$(\psi_{\mathbf k})$$ . A classical example is given by absolutely convergent series $$\sum_{\mathbf k\in\mathbb Z^d}\mathopen|a_{\mathbf k}|<\infty$$ 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 $$\Biggl(\,\sum_{\mathbf k\in G_j\setminus G_{j-1}}\mathopen|a_{\mathbf k}|^\beta\Biggr)^{\negthinspace 1/\beta} \le 2^{-rj} j^b, \qquad j\in\mathbb N,$$ where the index sets $$\mathcal G=(G_j)$$ satisfy $$G_{j-1}\subset G_j$$ and $$\bigcup_{j=1}^\infty G_j = \mathbb Z^d$$ . It has been 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 generalize these particular results to the classes of functions defined by index sets $$(G_j)$$ of a rather general structure.