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A continuity property of multivariate Lagrange interpolation

Authors: Thomas Bloom and Jean-Paul Calvi
Journal: Math. Comp. 66 (1997), 1561-1577
MSC (1991): Primary 41A05, 41A63
MathSciNet review: 1422785
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Abstract: Let $\{S_{t}\}$ be a sequence of interpolation schemes in ${\mathbb {R}}^{n}$ of degree $d$ (i.e. for each $S_{t}$ one has unique interpolation by a polynomial of total degree $\leq d)$ and total order $\leq l$. Suppose that the points of $S_{t}$ tend to $0 \in {\mathbb {R}}^{n}$ as $t \to \infty$ and the Lagrange-Hermite interpolants, $H_{S_{t}}$, satisfy $\lim _{t\to \infty } H_{S_{t}} (x^{\alpha }) = 0$ for all monomials $x^{\alpha }$ with $|\alpha | = d+1$. Theorem: $\lim _{t\to \infty } H_{S_{t}} (f) = T^{d} (f)$ for all functions $f$ of class $C^{l-1}$ in a neighborhood of $0$. (Here $T^{d} (f)$ denotes the Taylor series of $f$ at 0 to order $d$.) Specific examples are given to show the optimality of this result.

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Additional Information

Thomas Bloom
Affiliation: Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada

Jean-Paul Calvi
Affiliation: Laboratoire de mathématiques, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex, France

Keywords: Multivariable Lagrange interpolants, interpolation schemes in ${\mathbb {R}}^{n}$, Kergin interpolation
Received by editor(s): January 30, 1996
Received by editor(s) in revised form: August 21, 1996
Additional Notes: The first author was supported by NSERC of Canada.
Article copyright: © Copyright 1997 American Mathematical Society