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

DOI:
https://doi.org/10.1090/S0025-5718-97-00858-2

MathSciNet review:
1422785

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Abstract | References | Similar Articles | Additional Information

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.

- Thomas Bloom,
*Interpolation at discrete subsets of ${\bf C}^n$*, Indiana Univ. Math. J.**39**(1990), no. 4, 1223–1243. MR**1087190**, DOI https://doi.org/10.1512/iumj.1990.39.39055 - L. Bos,
*On certain configurations of points in ${\bf R}^n$ which are unisolvent for polynomial interpolation*, J. Approx. Theory**64**(1991), no. 3, 271–280. MR**1094439**, DOI https://doi.org/10.1016/0021-9045%2891%2990063-G - Jean Paul Calvi,
*Polynomial interpolation with prescribed analytic functionals*, J. Approx. Theory**75**(1993), no. 2, 136–156. MR**1249394**, DOI https://doi.org/10.1006/jath.1993.1094 - P. G. Ciarlet and P.-A. Raviart,
*General Lagrange and Hermite interpolation in ${\bf R}^{n}$ with applications to finite element methods*, Arch. Rational Mech. Anal.**46**(1972), 177–199. MR**336957**, DOI https://doi.org/10.1007/BF00252458 - Christian Coatmélec,
*Approximation et interpolation des fonctions différentiables de plusieurs variables*, Ann. Sci. École Norm. Sup. (3)**83**(1966), 271–341 (French). MR**0232143** - Lars Hörmander,
*An introduction to complex analysis in several variables*, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR**1045639** - Paul Kergin,
*A natural interpolation of $C^{K}$ functions*, J. Approx. Theory**29**(1980), no. 4, 278–293. MR**598722**, DOI https://doi.org/10.1016/0021-9045%2880%2990116-1 - S. L. Lee and G. M. Phillips,
*Interpolation on the triangle and simplex*, Approximation theory, wavelets and applications (Maratea, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, Kluwer Acad. Publ., Dordrecht, 1995, pp. 177–196. MR**1340890** - Rudolph A. Lorentz,
*Multivariate Birkhoff interpolation*, Lecture Notes in Mathematics, vol. 1516, Springer-Verlag, Berlin, 1992. MR**1222648** - Charles A. Micchelli,
*A constructive approach to Kergin interpolation in ${\bf R}^{k}$: multivariate $B$-splines and Lagrange interpolation*, Rocky Mountain J. Math.**10**(1980), no. 3, 485–497. MR**590212**, DOI https://doi.org/10.1216/RMJ-1980-10-3-485 - Günther Nürnberger,
*Approximation by spline functions*, Springer-Verlag, Berlin, 1989. MR**1022194** - Thomas Sauer and Yuan Xu,
*A case study in multivariate Lagrange interpolation*, Approximation theory, wavelets and applications (Maratea, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, Kluwer Acad. Publ., Dordrecht, 1995, pp. 443–452. MR**1340908** - 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

Email:
bloom@math.toronto.edu

**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