×

On multivariate Lagrange interpolation. (English) Zbl 0823.41002

Summary: Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree \(n\) of a function \(f\), which is a sum of integrals of certain \((n+ 1)\)st directional derivatives of \(f\) multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

MSC:

41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
41A63 Multidimensional problems
65D05 Numerical interpolation
65D10 Numerical smoothing, curve fitting
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carl de Boor and Amos Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp. 58 (1992), no. 198, 705 – 727. · Zbl 0767.41003
[2] M. Gasca, Multivariate polynomial interpolation, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 215 – 236. · Zbl 0694.41006 · doi:10.1098/rspa.1968.0185
[3] Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966.
[4] Rudolph A. Lorentz, Multivariate Birkhoff interpolation, Lecture Notes in Mathematics, vol. 1516, Springer-Verlag, Berlin, 1992. · Zbl 0760.41002
[5] Charles A. Micchelli, On a numerically efficient method for computing multivariate \?-splines, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 211 – 248. · Zbl 0422.41008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.