In this thesis we are concerned with the approximation of functions by radial basis function\udinterpolants. 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.\udThis 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.\udSuch estimates have application in the numerical analysis of solving partial differential equations using radial basis function collocation methods. At first our discussion focuses 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 applicability and a more satisfactory form. The method of proof employed is not restricted to interpolation alone. Rather, the technique provides error estimates for the approximation of rough functions for a variety of related approximation schemes as well.\udFor the previously mentioned class of radial basis functions, this work also gives error estimates when the function being interpolated has some additional smoothness. We find that the usual Lp-error estimate, for 1 ≤ p ≤ ∞, where the approximand belongs to the corresponding native space, can be doubled. Furthermore, error estimates are established for functions with smoothness intermediate to that of the native space and the subspace of the native space where double the error is observed.
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