Many researchers have discussed proximation by radial basis functions on a sphere, using scattered data. Usually there is no polynomial component in such approximations if, as here, the kernelthat 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.

The convergence theory (including a doubled rate of convergence for sufficiently smooth functions) make 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.