Good approximation on the sphere, with application to geodesy and the scattering of sound

Advances in approximation theory are often driven by applications. This paper explores two recent developments in polynomial approximation on the sphere, and relates them to applications. The first, driven by applications in geodesy, concerns cubature rules on the sphere that are exact for polynomials of high degree. The second, driven by a problem of the scattering of acoustic waves by smooth objects, concerns the so-called ‘hyperinterpolation’ approach to function approximation by spherical polynomials.