Error estimates for interpolation of rough and smooth functions using radial basis functions

In this thesis we are concerned with the approximation of functions by radial basis function interpolants. There is a plethora of results about the asymptotic behaviour of the error between appropriately smooth functions and their interpolants, as the interpolation points fill out a bounded domain in Euclidean space. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function—the native space. This work establishes Lp-error estimates, for 1 ≤ p ≤ ∞, when the function being interpolated fails to have the required smoothness to lie in the corresponding native space; therefore, providing error estimates for a class of rougher functions than previously known. Such estimates have application in the numerical analysis of solving partial differential equations using radial basis function collocation methods. At first our discussion focusses on the popular polyharmonic splines. A more general class of radial basis functions is admitted into exposition later on, this class being characterised by the algebraic decay of the Fourier transform of the radial basis function. The new estimates presented here offer some improvement on recent contributions from other authors by having wider applica