Multiscale analysis in Sobolev spaces on bounded domains

We study a multiscale scheme for the approximation of Sobolev functions on bounded domains. Our method employs scattered data sites and compactly supported radial basis functions of varying support radii at scattered data sites. The actual multiscale approximation is constructed by a sequence of residual corrections, where different support radii are employed to accommodate different scales. Convergence theorems for the scheme are proven, and it is shown that the condition numbers of the linear systems at each level are independent of the 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 on a bounded domain.