Note: It is not essential to have attended Part I.

Abstract: Many researchers have discussed approximation by radial basis functions on a sphere, using scattered data. Usually there is no polynomial component in such approximations if, as assumed here, the kernel that generates the radial functions is (strictly) positive definite.

On the other hand, the utility of polynomials for approximating slowly varying components is well known -- an extreme case is the NASA model of the earth's gravitational potential, which represents the potential by a purely
polynomial approximation of high degree.

In this joint work with Alvise Sommariva we
propose a hybrid approximation, in which there is a radial basis functions component to
handle the rapidly varying and localised aspects, but also a polynomial component to
handle the more slowly varying and global parts.

In this lecture I discuss the beautiful convergence theory (including a
doubled rate of convergence
for sufficiently smooth functions), which makes use of the ``native
space" associated with
the positive definite kernel (with no polynomial involvement in the
definition).
A numerical experiment for a simple model with a geophysical flavour
establishes
the potential value of the hybrid approach.