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

TL;DR

This paper develops a Green kernel approach using differential and boundary operators to explicitly construct reproducing kernels for Sobolev spaces, enhancing understanding of function approximation in these spaces.

Contribution

It introduces a novel method to derive reproducing kernels of Sobolev spaces via Green kernels associated with differential operators and boundary conditions, with explicit eigenfunction and eigenvalue characterizations.

Findings

Derived explicit Green kernels for Sobolev spaces

Established relationships between eigenfunctions and eigenvalues

Provided series expansions and orthonormal bases for the kernels

Abstract

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.