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In this dissertation, it is first shown that, when the radial basis functionis a $p$-norm and $1 < p < 2$, interpolation is always possible when the pointsare all different and there are at least two of them. We then show thatinterpolation is not always possible when $p > 2$. Specifically, for every $p >2$, we construct a set of different points in some $\Rd$ for which theinterpolation matrix is singular.  The greater part of this work investigates the sensitivity of radial basisfunction interpolants to changes in the function values at the interpolationpoints. Our early results show that it is possible to recast the work of Ball,Narcowich and Ward in the language of distributional Fourier transforms in anelegant way. We then use this language to study the interpolation matricesgenerated by subsets of regular grids. In particular, we are able to extend theclassical theory of Toeplitz operators to calculate sharp bounds on the spectraof such matrices.  Applying our understanding of these spectra, we construct preconditioners forthe conjugate gradient solution of the interpolation equations. Our main resultis that the number of steps required to achieve solution of the linear systemto within a required tolerance can be independent of the number ofinterpolation points. The Toeplitz structure allows us to use fast Fouriertransform techniques, which imp lies that the total number of operations is amultiple of $n \log n$, where $n$ is the number of interpolation points.  Finally, we use some of our methods to study the behaviour of themultiquadric when its shape parameter increases to infinity. We find asurprising link with the {\it sinus cardinalis} or {\it sinc} function ofWhittaker. Consequently, it can be highly useful to use a large shape parameterwhen approximating band-limited functions.

The Interpolation Theory of Radial Basis Functions