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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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  • [B] T. Bloom, Interpolation at discrete subsets of $\mathbb {C}^{n}$, Indiana J. of Math. 39 (1990), 1223-1243. MR 91k:32015
  • [Bo] L. Bos, On certain configurations of points in $\mathbb {R}^{n}$ which are unisolvent for polynomial interpolation, J. of Approx. Theory 64 (1991), 271-280. MR 91m:41005
  • [C] J. P. Calvi, Polynomial interpolation with prescribed analytic functionals, J. Approx. Theory 75 (1993), 136-156. MR 94j:41002
  • [CR] P.G. Ciarlet and P.A. Raviart, General Lagrange and Hermite interpolation in $\mathbb {R}^{n}$ with applications to finite element methods, Arch. Rat. Mech. Anal. 46 (1972), 177-199. MR 49:1730
  • [Co] C. Coatmelec, Approximation et interpolation des fonctions differentiables de plusieurs variables, Ann. Scient. Ec. Norm. Sup. 83 (1966), 271-341. MR 38:469
  • [H] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North Holland, Amsterdam, 1990. MR 91a:32001
  • [K] P. Kergin, A natural interpolation of $C^{k}$ functions, J. of Approx. 29 no 4 (1980), 278-293. MR 82b:41007
  • [LP] S. L. Lee and G. M. Phillips, Interpolation on the triangle and simplex, Approximation Theory, Wavelets and Applications (S. P. Singh, ed.), Kluwer Academic Publishers, 1995, pp. 177-196. MR 96f:41042
  • [L] R. A. Lorentz, Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics no. 1516, Springer-Verlag. MR 94h:41001
  • [M] C. A. Micchelli, A constructive approach to Kergin interpolant in $\mathbb {R}^{k}$, Rocky Mountain J. 10 (3) (1980), 485-497. MR 84i:41002
  • [N] G. Nürnberger, Approximation by spline functions, Springer, Berlin, 1989. MR 90j:41025
  • [SX] T. Sauer and Y. Xu, A case study in multivariate Lagrange interpolation, Approximation Theory, Wavelets and Applications (S. P. Singh, ed.), Kluwer Academic Publishers, 1995, pp. 443-452. MR 96d:41036
  • [W] S. Waldron, Integral error formula for the scale of mean value interpolations which includes Kergin and Hakopian interpolation, Numer. Math. (to appear).

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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

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