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The $L_{2}$-approximation order of surface spline interpolation

Author: Michael J. Johnson
Journal: Math. Comp. 70 (2001), 719-737
MSC (2000): Primary 41A15, 41A25, 41A63, 65D05
Published electronically: October 27, 2000
MathSciNet review: 1813145
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Abstract: We show that if the open, bounded domain $\Omega \subset \mathbb{R}^{d}$ has a sufficiently smooth boundary and if the data function $f$ is sufficiently smooth, then the $L_{p}(\Omega )$-norm of the error between $f$ and its surface spline interpolant is $O(\delta ^{\gamma _{p}+1/2})$ ( $1\leq p\leq \infty $), where $\gamma _{p}:=\min \{m,m-d/2+d/p\}$ and $m$ is an integer parameter specifying the surface spline. In case $p=2$, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the $L_{2}$-approximation order of surface spline interpolation is $m+1/2$.

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Additional Information

Michael J. Johnson
Affiliation: Deptartment of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait

Keywords: Interpolation, surface spline, approximation order, scattered data
Received by editor(s): June 10, 1999
Published electronically: October 27, 2000
Additional Notes: This work was supported by Kuwait University Research Grant SM-175.
Article copyright: © Copyright 2000 American Mathematical Society

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