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 Ψ=(ψₖ) , i.e., f = ∑k∈ Zᵈ aₖ ψₖ , with certain structural conditions imposed on the coefficients (aₖ) and certain conditions imposed on the basis (ψₖ) . A classical example is given by absolutely convergent series ∑k∈ Zᵈ|aₖ|<∞ with respect to an orthogonal or a Riesz basis (ψₖ) , or even a redundant set of functions. Here, we study the classes of functions A_βr,b(Ψ, G) with the property (\,∑_ k∈ Gⱼ Gⱼ₋₁|aₖ|^β)1/β ≤ 2⁻ʳʲ jᵇ, j∈ N, where the index sets G=(Gⱼ) satisfy Gⱼ₋₁⊂ Gⱼ and ⱼ₌₁^∞ Gⱼ = Zᵈ . 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 (Gⱼ) are dyadic cubes or dyadic hyperbolic crosses. In this paper, we generalize these particular results to the classes of functions defined by index sets (Gⱼ) of a rather general structure.