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A continuity property of multivariate Lagrange interpolation

Authors:	Thomas Bloom Jean-Paul Calvi

Institution:	Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada ; Laboratoire de mathématiques, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex, France

Abstract:	Let $\{S_{t}\}$ be a sequence of interpolation schemes in ${\mathbb {R}}^{n}$ of degree (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 of class $C^{l-1}$ in a neighborhood of . (Here $T^{d} (f)$ denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

Keywords:	Multivariable Lagrange interpolants interpolation schemes in ${\mathbb{R}}^{n}$ Kergin interpolation

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