The Lp-Approximation Order of Surface Spline Interpolation for 1 \leq p \leq 2

We show that the Lp-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + ½ and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain Ω, we show specifically that the Lp(Ω)-norm of the error between f and its surface spline interpolant is O(hm + 1/p) provided that f belongs to an appropriate Sobolev or Besov space and that Ω \subset Rd is open, bounded, and has the C2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the Lp-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.