Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators

Details Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators

2011-09-27

arxiv.org/abs/1109.5755v1

Mathematics

Numerical Analysis

Scientific paper

We introduce a vector differential operator $\mathbf{P}$ and a vector boundary operator $\mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator $L:=\mathbf{P}^{\ast T}\mathbf{P}$ with homogeneous or nonhomogeneous boundary conditions given by $\mathbf{B}$, where we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators $\mathbf{P}$ and $\mathbf{B}$. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.

Affiliated with

Also associated with

No associations

Rating

Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-61802