Towards a theory of multivariate interpolation using spaces of distributions.

The research contained in this thesis concerns the study of multivariate interpolation problems. Given a discrete set of possibly complex-valued data, indexed by a set of interpolation nodes in Euclidean space, it is desirable to generate a function which agrees with the data at the nodes. Within this general framework, this work pursues and generalizes one approach to the problem. Based on a variational theory, we construct a parameterised family of Hilbert spaces of tempered distributions, detail the necessary evolution of the interpolation problem, and provide a general error analysis. Some of the more popular applications from the theory of radial basis functions are shown to arise naturally, but the theory admits many more examples, which are not necessarily radial. The general error analysis is applied to each of the applications, and taken further where possible. Connections with the theory of conditionally positive definite functions are highlighted, but are not central to the theme.