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In this paper we introduce a generalization of the classical$\Leb_2(\Rd)$-based Sobolev spaces with the help of a vector differentialoperator $\mathbf{P}$ which consists of finitely or countably many differentialoperators $P_n$ which themselves are linear combinations of distributionalderivatives. We find that certain proper full-space Green functions $G$ withrespect 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 distributionalsense. We then provide sufficient conditions under which our generalizedSobolev space will become a reproducing-kernel Hilbert space whose reproducingkernel can be computed via the associated Green function $G$. As an applicationof this theoretical framework we use $G$ to construct multivariate minimum-norminterpolants $s_{f,X}$ to data sampled from a generalized Sobolev function $f$on $X$. Among other examples we show the reproducing-kernel Hilbert space ofthe Gaussian function is equivalent to a generalized Sobolev space.

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