Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [13] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of IRd and spheres. As a step on the way to our ultimate goal we also show convergence of pseudo-derivatives of the interpolation error.