The Error in Polynomial Tensor-Product, and Chung-Yao, Interpolation

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 formulæ 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 formulæ, 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 formulæ. 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 formulæ 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 Méhauté, 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.