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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computational aspects of polynomial interpolation in several variables

Authors: Carl de Boor and Amos Ron
Journal: Math. Comp. 58 (1992), 705-727
MSC: Primary 65D05; Secondary 41A05, 41A63
MathSciNet review: 1122061
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Abstract: The pair $ \langle \Theta ,P\rangle $ of a point set $ \Theta \subset {\mathbb{R}^d}$ and a polynomial space P on $ {\mathbb{R}^d}$ is correct if the restriction map $ P \to {\mathbb{R}^\Theta }:p \mapsto {p_{\vert\Theta }}$ is invertible, i.e., if there is, for any f defined on $ \Theta $, a unique $ p \in P$ which matches f on $ \Theta $.

We discuss here a particular assignment $ \Theta \mapsto {\Pi _\Theta }$, introduced by us previously, for which $ \langle \Theta ,{\Pi _\Theta }\rangle $ is always correct, and provide an algorithm for the construction of a basis for $ {\Pi _\Theta }$, which is related to Gauss elimination applied to the Vandermonde matrix $ {({\vartheta ^\alpha })_{\vartheta \in \Theta ,\alpha \in \mathbb{Z}_ + ^d}}$ for $ \Theta $. We also discuss some attractive properties of the above assignment and algorithmic details, and present some bivariate examples.

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Keywords: Exponentials, polynomials, multivariate, interpolation, multivariable Vandermonde, Gauss elimination, harmonic polynomials
Article copyright: © Copyright 1992 American Mathematical Society