On the Semi-norm of Radial Basis Function Interpolants

Radial basis function interpolation has attracted a lot of interest in recent years. For popular choices, for example thin plate splines, this problem has a variational formulation, i.e. the interpolant minimizes a semi-norm on a certain space of radial functions. This gives rise to a function space, called the native space. Every function in this space has the property that the semi-norm of an arbitrary interpolant to this function is uniformly bounded. In applications it is of interest whether a suuciently smooth function belongs to the native space. In this paper we give suucient conditions on the diierentiability of a function with compact support, in the case of cubic, linear and thin plate splines. In the case of multi-quadrics and Gaussian functions, it is shown that the only compactly supported function that satisses these conditions is identically zero.