The Error in Polynomial Tensor-Product, and Chung-Yao, Interpolation
The Error in Polynomial Tensor-Product, and Chung-Yao, Interpolation
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发表时间：
2009
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通讯作者：
C. Boor
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作者：
C. Boor
本文用归纳法证明了钟耀插值法的误差公式。在这个过程中，证明了一个二元独立兴趣的除差恒等式。并给出了多项式张量积插值的一个误差公式的归纳证明。主要工具是一个（方便的符号）多元除差。文[2]中引入了一种特殊的多元差除，并给出了多元多项式插值的三种特殊情况的误差公式，但未证明。可以肯定的是，一个归纳程序表明，所以它声称有，将产生每一个这些公式，但（显然是至关重要的）细节是失踪，无论是在张量积插值和钟耀插值的错误。本文的目的是提供这些公式的完整的归纳证明，如[2]中所概述的。在证明钟耀插值公式的过程中，证明了一个本质上二元的除差恒等式，该恒等式具有很好的多元推广性。简短的“直接”证明（与归纳被隐藏在著名的结果单变量除差异）出现在[1]。本说明的结构简单如下。在第一节中，我们简要回顾了文[2]中引入的差除（记法），在第二节中，我们简要回顾了关于IR中一般位置的超平面的一些著名事实，在第三节中，我们给出了钟耀插值的误差公式的归纳证明，并在此过程中证明了两个有用的差除恒等式。这是其次是一个归纳证明的误差公式多项式张量积插值，在第4节。最后一节指出了这两个误差公式之间的相似之处，并推测了多项式插值在任意点集上的逐点误差公式的形式，特别注意了多项式插值在任意点集上的误差的Sauer-Xu公式，在该点集上，从次数≤ k的多项式的全空间Rk k的插值是唯一可能的。曲面拟合和多分辨率方法35 A.勒梅奥特角Rabut和L. L. Schumaker（eds.），pp. 35-50.范德比尔特大学出版社，纳什维尔，田纳西州版权所有。ISBN 0-8265-1294-1。保留一切形式的复制权。
A formula for the error in Chung-Yao interpolation announced earlier is proved (by induction). In the process, a bivariate divided difference identity of independent interest is proved. Also, an inductive proof of an error formula for polynomial interpolation by tensorproducts is given. The main tool is a (convenient notation for a) multivariate divided difference. In [2], a particular multivariate divided difference is introduced and, as an illustration of its usefulness, error formulae for three special cases of multivariate polynomial interpolation are stated, but not proved. To be sure, an inductive procedure is indicated which, so it is claimed there, will produce each of these formulae, but (apparently crucial) detail is missing, for both the error in tensor-product interpolation and in Chung-Yao interpolation. It is the purpose of this note to provide complete, inductive proofs, as outlined in [2], of these formulae. In the process of proving the formula for Chung-Yao interpolation, an essentially bivariate divided difference identity is proved which may well have a nice multivariate generalization. Short ‘direct’ proofs (with the induction being hidden in well-known results about univariate divided differences) appear in [1]. This note has the following simple structure. After a quick recall, in Section 1, of the divided difference (notation) introduced in [2], and, in Section 2, of well-known facts about hyperplanes in IR in general position, Section 3 brings the inductive proof of the error formula for Chung-Yao interpolation, proving two useful divided difference identities in the process. This is followed by an inductive proof of the error formula for polynomial tensor-product interpolation, in Section 4. The last section points out similarities between these two error formulae and speculates on the form of a pointwise error formula for polynomial interpolation at an arbitrary pointset, with particular attention to the Sauer-Xu formula for the error in polynomial interpolation at an otherwise arbitrary pointset at which interpolation from the full space Πk of polynomials of degree ≤ k is uniquely possible. Surface Fitting and Multiresolution Methods 35 A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds.), pp. 35–50. Copyright oc 1997 by Vanderbilt University Press, Nashville, TN. ISBN 0-8265-1294-1. All rights of reproduction in any form reserved.