A continuity property of multivariate Lagrange interpolation
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- by Thomas Bloom and Jean-Paul Calvi PDF
- Math. Comp. 66 (1997), 1561-1577 Request permission
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.References
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Additional Information
- Thomas Bloom
- Affiliation: Department of Mathematics, University of Toronto, M5S 1A1, Toronto, Ontario, Canada
- Email: bloom@math.toronto.edu
- Jean-Paul Calvi
- Affiliation: Laboratoire de mathématiques, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex, France
- 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.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 1561-1577
- MSC (1991): Primary 41A05, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-97-00858-2
- MathSciNet review: 1422785