Multiscale Approximation and Reproducing Kernel Hilbert Space Methods

We consider reproducing kernels K : ⌦ ⇥ ⌦ ! R in multiscale series expansion form, i.e., kernels of the form K (x, y) = P ` 2N`P j2I`` ,j (x) `,j (y) with weightsànd structurally simple basis functions`,i. Here, we deal with basis functions such as polynomials or frame systems, where, for`2 N, the index set I ` is finite or countable. We derive relations between approximation properties of spaces based on basis functions { ` (y) is provided where the truncation index L is chosen sufficiently large depending on the point set X N. Furthermore, Bernstein-type inverse estimates and derivative-free sampling inequalities for kernel based spaces are obtained from estimates for spaces based on the basis functions { `,j : 1  `  L, j 2 I ` }. 1. Introduction. In many applications, one seeks to reconstruct or to approximate a function from given (measured) strong or weak data. Kernel-based methods have proven a reliable and useful tool in a large variety of such problems, including meshless methods for the solution of partial differential equations, surface reconstruction and machine learning algorithms (see [4, 5, 6, 7, 24, 25, 28]). In kernel-based meshless methods one typically considers trial spaces that are based on a finite discrete set X N = {x 1 ,. .. , x N } ⇢ ⌦ of data points. They are spanned by translates of a given kernel function