Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

The theories for radial basis functions (RBFs) as well as piecewise polynomial splines have reached a stage of relative maturity as is demonstrated by the recent publication of a number of monographs in either field. However, there remain a number of issues that deserve to be investigated further. For instance, it is well known that both splines and radial basis functions yield “optimal” interpolants, which in the case of radial basis functions are discussed within the so-called native space setting. It is also known that the theory of reproducing kernels provides a common framework for the interpretation of both RBFs and splines. However, the associated reproducing kernel Hilbert spaces (or native spaces) are often not that well understood — especially in the case of radial basis functions. By linking (conditionally) positive definite kernels to Green’s functions of differential operators we obtain new insights that enable us to better understand the nature of the native space as a generalized Sobolev space. An additional feature that appears when viewing things from this perspective is the notion of scale built into the definition of these function spaces. Furthermore, the eigenfunction expansion of a positive definite kernel via Mercer’s theorem provides a tool for making progress on such important questions as stable computation with flat radial basis functions and dimension independent error bounds.