Polynomial Approximation on the Sphere, with Two Applications
by
Ian H. Sloan
University of New South Wales, Australia

The theme of this talk is that polynomial approximations on the sphere are important for applications, but that successful applications of high-degree polynomials need a good understanding of underlying approximation properties. We illustrate with two case studies.

First, for applications in geodesy, there is good reason to use cubature rules that have a high degree of polynomial accuracy. The stability, and even the computability, of such rules depends critically on the properties of the underlying polynomial interpolants. Second, a recent spectral approach to the scattering of sound by three-dimensional objects needs for its analysis good approximation properties of the 'hyperinterpolation' polynomial approximation scheme.

In the course of this talk the existing state of knowledge for both interpolation and hyperinterpolation will be reviewed.

Date received: February 6, 2002