Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators |
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Authors: | Gregory E Fasshauer Qi Ye |
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Institution: | 1.Department of Applied Mathematics,Illinois Institute of Technology,Chicago,USA |
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Abstract: | In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional
operator P consisting of finitely or countably many distributional operators P
n
, which are defined on the dual space of the Schwartz space. The types of operators we 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 function G with respect to L := P
*T
P 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 embedded 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 ì \mathbbRd{X \subset \mathbb{R}^d}. We 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. These examples further illustrate
how we can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the “best”
kernel function for kernel-based approximation methods. |
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Keywords: | |
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