On a polyharmonic Dirichlet problem and boundary effects in surface spline approximation

For compact domains with smooth boundaries, we present an approximation scheme for surface spline approximation that delivers precise $L_p$ approximation orders on well known smoothness spaces. This scheme overcomes the boundary effects when centers are placed with greater density near to the boundary. It owes its success to an integral identity using a minimal number of boundary layer potentials, which, in turn is derived from the boundary layer potential solution to the Dirichlet problem for the $m$-fold Laplacian. Furthermore, his integral identity is shown to be the "native space extension" of the target function.