Abstract
GC-sets are subsets T of \(\mathbb{R}^d\) of the appropriate cardinality \(\dim\Pi_n\) for which, for each τ ∈ T, there are n hyperplanes whose union contains all of T except for τ, thus making interpolation to arbitrary data on T by polynomials of degree ≤ n uniquely possible. The existing bivariate theory of such sets is extended to the general multivariate case and the concept of a maximal hyperplane for T is highlighted, in hopes of getting more insight into existing conjectures for the bivariate case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berzolari, L.: Sulla determinazione d’una curva o d’una superficie algebrica e su alcune questioni di postulazione. Rendiconti del R. Instituto Lombardo Ser. (2) 47, 556–564 (1914)
de Boor, C.: Gauss elimination by segments and multivariate polynomial interpolation. In: Zahar, R.V.M. (ed.) Approximation and Computation: A Festschrift in Honor of Walter Gautschi, pp. 1–22. ISNM 119, Birkhäuser Verlag, Basel-Boston-Berlin (1994)
de Boor, C., Ron, A.: On multivariate polynomial interpolation. Constr. Approx. 6, 287–302 (1990)
de Boor, C., Ron, A.: Computational aspects of polynomial interpolation in several variables. Math. Comput. 58(198), 705–727 (1992)
Bos, L.: On certain configurations of points in \(\mathbb{R}^n\) which are unisolvent for polynomial interpolation. J. Approx. Theory 64, 271–280 (1991)
Bos, L., Caliari, M., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143(1), 15–25 (2006)
Busch, J.R.: A note on Lagrange interpolation in \(\mathbb{R}^2\). Rev. Union Mat. Argent. 36, 33–38 (1990)
Carnicer, J.M., Gasca, M.: Planar configurations with simple Lagrange formula. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces III, Oslo, 2000, pp. 55–62. Vanderbilt University Press, Nashville (2001)
Carnicer, J., Gasca, M.: A conjecture on multivariate polynomial interpolation. Rev. R. Acad. Cienc. Ser. A Mat. 95, 145–153 (2001)
Carnicer, J.M., Gasca, M.: On Chung and Yao’s geometric characterization for bivariate polynomial interpolation. In: Lyche, T., Mazure, M.-L., Schumaker, L.L. (eds.) Curve and Surface Design: Saint-Malo 2002, pp. 11–30. Nashboro Press, Brentwood, TN (2003)
Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14, 735–743 (1977)
Coatmélec, C.: Approximation et interpolation des fonctions différentiables de plusieurs variables. Ann. Sci. École Norm. Sup. 83(3), 271–341 (1966)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Berlin Heidelberg New York (1992)
Gasca, M., Maeztu, J.I.: On Lagrange and Hermite interpolation in \(\mathbb{R}^k\). Numer. Math. 39, 1–14 (1982)
Liang, X.-Z., Lü, C.-M., Feng, R.Z.: Properly posed sets of nodes for multivariate Lagrange interpolation in C s. SIAM J. Numer. Anal. 39(2), 587–595 (2001–2002)
Nicolaides, R.A.: On a class of finite elements generated by Lagrange interpolation. SIAM J. Numer. Anal. 9, 435–445 (1972)
Radon, J.: Zur mechanischen Kubatur. Monatshefte der Math. Physik 52(4), 286–300 (1948)
Sauer, T.: Some more facts about CG n . note from 13jan (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Boor, C. Multivariate polynomial interpolation: conjectures concerning GC-sets. Numer Algor 45, 113–125 (2007). https://doi.org/10.1007/s11075-006-9062-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-006-9062-2
Keywords
- Gasca–Maetzu conjecture
- Geometric characterization
- Completely factorizable
- Lagrange form
- Maximal polynomial