This paper studies the approximation of a general d-variate function f by shifts of a fixed function φ, where the centers Ξ are arbitrary. The authors establish approximation theorems whose error bounds reflect the local density of the points in Ξ. Two settings are analyzed: one where Ξ is fixed, and another where Ξ can be chosen based on the target function f. In the first setting, improved approximation occurs in regions with high density. In the second setting, the authors consider non-linear approximation, where Ξ is chosen to minimize the approximation error. They show that a function can be approximated in L_p(R^d) with error O(N^{-s/d}) if it lies in the Triebel-Lizorkin space F_{τ,q}^s(R^d), where s, p, and τ are related by 1/τ - 1/p = s/d and q = (1 + s/d)^{-1}. The authors also derive corresponding theorems for N-term approximation in terms of Besov classes. The paper discusses the use of wavelet decompositions to characterize smoothness spaces and provides error estimates for approximation in both linear and non-linear settings. The results are applied to functions with lower smoothness, where the approximation error is controlled by decomposing the function into a smooth part and a nonsmooth part. The authors also discuss the use of scattered data and radial basis functions in approximation problems.This paper studies the approximation of a general d-variate function f by shifts of a fixed function φ, where the centers Ξ are arbitrary. The authors establish approximation theorems whose error bounds reflect the local density of the points in Ξ. Two settings are analyzed: one where Ξ is fixed, and another where Ξ can be chosen based on the target function f. In the first setting, improved approximation occurs in regions with high density. In the second setting, the authors consider non-linear approximation, where Ξ is chosen to minimize the approximation error. They show that a function can be approximated in L_p(R^d) with error O(N^{-s/d}) if it lies in the Triebel-Lizorkin space F_{τ,q}^s(R^d), where s, p, and τ are related by 1/τ - 1/p = s/d and q = (1 + s/d)^{-1}. The authors also derive corresponding theorems for N-term approximation in terms of Besov classes. The paper discusses the use of wavelet decompositions to characterize smoothness spaces and provides error estimates for approximation in both linear and non-linear settings. The results are applied to functions with lower smoothness, where the approximation error is controlled by decomposing the function into a smooth part and a nonsmooth part. The authors also discuss the use of scattered data and radial basis functions in approximation problems.