  In this dissertation, it is first shown that, when the radial basis function
is a $p$-norm and $1 < p < 2$, interpolation is always possible when the points
are all different and there are at least two of them. We then show that
interpolation is not always possible when $p > 2$. Specifically, for every $p >
2$, we construct a set of different points in some $\Rd$ for which the
interpolation matrix is singular.
  The greater part of this work investigates the sensitivity of radial basis
function interpolants to changes in the function values at the interpolation
points. 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 an
elegant way. We then use this language to study the interpolation matrices
generated by subsets of regular grids. In particular, we are able to extend the
classical theory of Toeplitz operators to calculate sharp bounds on the spectra
of such matrices.
  Applying our understanding of these spectra, we construct preconditioners for
the conjugate gradient solution of the interpolation equations. Our main result
is that the number of steps required to achieve solution of the linear system
to within a required tolerance can be independent of the number of
interpolation points. The Toeplitz structure allows us to use fast Fourier
transform techniques, which imp lies that the total number of operations is a
multiple of $n \log n$, where $n$ is the number of interpolation points.
  Finally, we use some of our methods to study the behaviour of the
multiquadric when its shape parameter increases to infinity. We find a
surprising link with the {\it sinus cardinalis} or {\it sinc} function of
Whittaker. Consequently, it can be highly useful to use a large shape parameter
when approximating band-limited functions.
