Multiscale Analysis in Sobolev Spaces on the Sphere

We consider a multiscale approximation scheme at scattered sites for functions in Sobolev spaces on the unit sphere Sⁿ. The approximation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to Sⁿ. A convergence theorem for the scheme is proved, and the condition number of the linear system is shown to stay bounded by a constant from level to level, thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points.