Error Estimates for Thin Plate Spline Approximation in the Disc

This paper is concerned with approximation properties of linear combinations of scattered translates of the thin-plate spline radial basis function | · | log | · | where the translates are taken in the unit disc D in R. We show that the Lp approximation order for this kind of approximation is 2+1/p (for sufficiently smooth functions), which matches Johnson’s upper bound and, thus, gives the saturation order. We also show that when one increases the density of the centers at the boundary, approximation order 4 – the best possible order in the absence of a boundary – can be obtained. AMS classification: 41A15, 41A25, 41A63.