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Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators | | Qi Ye
; | | Date: |
1 Sep 2011 | | Abstract: | In this paper we introduce a generalization of the classical
$Leb_2(Rd)$-based Sobolev spaces with the help of a vector differential
operator $mathbf{P}$ which consists of finitely or countably many differential
operators $P_n$ which themselves are linear combinations of distributional
derivatives. We find that certain proper full-space Green functions $G$ with
respect to $L=mathbf{P}^{ast T}mathbf{P}$ are positive definite functions.
Here we ensure that the vector distributional adjoint operator
$mathbf{P}^{ast}$ of $mathbf{P}$ is well-defined in the distributional
sense. We then provide sufficient conditions under which our generalized
Sobolev space will become a reproducing-kernel Hilbert space whose reproducing
kernel can be computed via the associated Green function $G$. As an application
of this theoretical framework we use $G$ to construct multivariate minimum-norm
interpolants $s_{f,X}$ to data sampled from a generalized Sobolev function $f$
on $X$. Among other examples we show the reproducing-kernel Hilbert space of
the Gaussian function is equivalent to a generalized Sobolev space. | | Source: | arXiv, 1109.0109 | | Services: | Forum | Review | PDF | Favorites | |
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