Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators

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.