PDLg splines defined by partial differential operators with initial and boundary value conditions
SIAM Journal on Numerical Analysis
Radial Basis Functions
Convergence of Unsymmetric Kernel-Based Meshless Collocation Methods
SIAM Journal on Numerical Analysis
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
Support Vector Machines
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Multivariate interpolation with increasingly flat radial basis functions of finite smoothness
Advances in Computational Mathematics
Learning with boundary conditions
Neural Computation
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We introduce a vector differential operator P and a vector boundary operator 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:驴=驴P 驴驴驴T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P 驴驴驴 of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and 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.