  In this paper we consider 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 R^d. 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. In many
cases, the native space contains functions possessing a certain amount of
smoothness. We address the question of what can be said about these error
estimates when the function being interpolated fails to have the required
smoothness. These are the rough functions of the title. We limit our discussion
to surface splines, as an exemplar of a wider class of radial basis functions,
because we feel our techniques are most easily seen and understood in this
setting.
