Multiscale analysis in Sobolev spaces on bounded domains with zero boundary values

Compactly supported radial basis functions can be used to define multiscale approximation spaces. A multiscale scheme is studied for the approximation of Sobolev functions on bounded domains with restricted data sites. Our method employs compactly supported radial basis functions with centres at scattered data sites chosen at each level to ensure the support of the interpolant is contained within the domain. The multiscale approximation is constructed by successive residual corrections at each level using support radii appropriate to capture different scales. Convergence for the scheme is proven and numerical examples are given.