Native Hilbert Spaces for Radial Basis Functions I

Abstract

This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive definite case. The definitions of the latter are based on finitely supported functionals only. Fourier or other transforms are not required. The dependence of native spaces on the domain is studied, and criteria for functions and functionals to be in the native space are given. Basic facts on optimal recovery, power functions, and error bounds are included.

Interpolation of spatial data – A stochastic or a deterministic problem?

“…Then, by the theory of reproducing kernel Hilbert spaces (see [66] and references therein), a unique symmetric function K : T × T → R ("reproducing kernel") exists with K(·, x) ∈ H and…”

“…We chose it because of its striking analogy with the theory of intrinsic random fields [52], which will be further discussed in section 3. In [66] a detailed derivation of the native space of a given conditionally positive definite kernel K is presented, showing how values g(x), x ∈ T can be assigned to the abstract function g which, a priori, can be evaluated only by functionals from L P (T ) which does not include the point evaluation functionals δ x . Note that the positive definite case discussed above corresponds to P = {0}, and continuity of any λ ∈ L {0} (T ) is a consequence of assumption (1) and the Riesz representation theorem.…”

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

“…Since the kernel K is positive definite, it is the reproducing kernel of its native Hilbert space [13,14] defined as the space of all generalized functions f on IR d with…”

“…by Lemma 3 for µ and ρ restricted by (13). This implies that one can solve the system (15) by some trial function u * r,s of the form (14) approximately to some accuracy…”

Error bounds for kernel-based numerical differentiation

“…Going over to the completion turns the kernel part into a Hilbert space, and thus there is a native space F K of functions on Ω which is the direct sum of a Hilbert space and the polynomial space Π d s , arising as completion of the space of functions of the form (6). It is not straightforward to interpret the abstract completion as a space of functions, but we leave details on this to [9].…”