MULTISCALE APPROXIMATION AND REPRODUCING KERNEL HILBERT SPACE METHODS

摘要

We consider reproducing kernels K : Omega x Omega -> R in multiscale series expansion form, i.e., kernels of the form K (x, y) = Sigma(l is an element of N) lambda(l) Sigma(j is an element of Il) phi(l,j) (x) phi(l,j) (y) with weights lambda(l) and structurally simple basis functions {phi(l,j)}. Here, we deal with basis functions such as polynomials or frame systems, where, for l is an element of N, the index set I is finite or countable. We derive relations between approximation properties of spaces based on basis functions {phi(l,j) : 1 <= l <= L, j is an element of I-l} and spaces spanned by translates of the kernel span {K(x(1),.),...,K(x(N),.)} with X-N := {x(1),...,x(N)} subset of Omega if the truncation index L is appropriately coupled to the discrete set X-N. An analysis of a numerically feasible approximation from trial spaces span {K-L(x(1),.),...,K-L(x(N),.)} based on finitely truncated series kernels of the form K-L (x, y) := Sigma(L)(l=1) lambda(l) Sigma(j is an element of Il), phi(l,j) (x) phi(l,j) (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 {phi(l,j) : 1 <= l <= L,j is an element of I-l}.