Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

In this paper we extend the definition of generalized Sobolev space and subsequent theoretical results established recently for positive definite kernels and differential operators in the article [21]. In the present paper the semi-inner product of the generalized Sobolev space is set up by a vector distributional operator P consisting of finitely or countably many distributional operators Pn, which are defined on the dual space of the Schwartz space. The types of operators we now consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green functionG with respect to L := P∗TP now becomes a conditionally positive function. In order to support this claim we ensure that the distributional adjoint operator P∗ of P is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be imbedded into or even be equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s f ,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set X ⊂ Rd. We will provide several examples, such as Matérn kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are equivalent to a generalized Sobolev space.