Abstract
We show that the L p -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 L p (Ω)-norm of the error between f and its surface spline interpolant is O(h m + 1/p) provided that f belongs to an appropriate Sobolev or Besov space and that Ω \subset R d is open, bounded, and has the C 2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h 2m)) 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 L p -approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.
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Johnson, M. The L p -Approximation Order of Surface Spline Interpolation for 1 \leq p \leq 2. Constr Approx 20, 303–324 (2004). https://doi.org/10.1007/s00365-003-0534-5
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DOI: https://doi.org/10.1007/s00365-003-0534-5