Abstract
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.
