The Lp-Approximation Order of Surface Spline Interpolation for 1 \leq p \leq 2
The Lp-Approximation Order of Surface Spline Interpolation for 1 \leq p \leq 2
复制标题
1 leq p leq 2 的曲面样条插值的 Lp 近似阶
DOI：
10.1007/s00365-003-0534-5
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发表时间：
2004
影响因子：
2.7
通讯作者：
Michael J. Johnson
中科院分区：
文献类型：
--
作者：
Michael J. Johnson
摘要
我们证明了曲面样条插值的lp逼近阶
等于m+1/p， p在1 \leq p \leq 2范围内，其中m是整数
参数，用于指定曲面样条。在此之前，人们知道
秩序的下限是m+1/ 2，上限是m+1/p。与
H表示插值点与区域之间的填充距离
Ω，我们具体地表明，Lp（Ω）-范数的误差之间的f
其表面样条插值为0 (hm + 1/p)，设f为
到合适的Sobolev或Besov空间Ω \subset
Rd是开的，有界的，具有c2m正则性
财产。我们还表明，边界效应（导致的速率）
收敛性明显差于O(h2m))，限制于a
其宽度不大于的常数倍的边界层
H |log H |。最后，给出了支持该理论的数值证据
推测
曲面样条插值的lp逼近阶数为m + 1/p
2 < p \leq\infty。
Abstract
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.