Abstract
The geometric characterization identifies the sets of nodes such that the Lagrange polynomials are products of factors of first degree. We offer a detailed classification of all known sets satisfying the geometric characterization in the plane. The defect, which takes into account the number of lines containing more nodes than the degree, plays a fundamental role in this classification.
Similar content being viewed by others
References
de Boor, C.: Multivariate polynomial interpolation: conjectures concerning GC-sets. Numer. Algorithms 45, 113–125 (2007)
Busch, J.R.: A note on Lagrange interpolation in R 2. Rev. Union Mat. Argent. 36, 33–38 (1990)
Carnicer, J.M., García-Esnaola, M.: Lagrange interpolation on conics and cubics. Comput. Aided Geom. Des. 19, 313–326 (2002)
Carnicer, J.M., Gasca, M.: Planar configurations with simple Lagrange formulae. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in CAGD: Oslo 2000, pp. 55–62. Vanderbilt University Press, Nashville (2001)
Carnicer, J.M., 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. 21–30. Nashboro, Brentwood (2003)
Carnicer, J.M., Gasca, M.: Classification of bivariate GC configurations for interpolation. Adv. Comput. Math. 20, 5–16 (2004)
Carnicer, J.M., Gasca, M.: Generation of lattices of points for bivariate interpolation. Numer. Algorithms 39, 69–79 (2005)
Carnicer, J.M., Gasca, M.: Interpolation on lattices generated by cubic pencils. Adv. Comput. Math. 24, 113–130 (2006)
Carnicer, J.M., Godés, C.: Geometric characterization and generalized principal lattices. J. Approx. Theory 143, 2–14 (2006)
Carnicer, J.M., Godés, C.: Geometric characterization of configurations with defect three. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curves and Surfaces Fitting: Avignon 2006, pp. 61–70. Nashboro, Brentwood (2007)
Carnicer, J.M., Godés, C.: Generalized principal lattices and cubic pencils. Numer. Algorithms 44, 133–145 (2007)
Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14, 735–743 (1977)
Gasca, M., Maeztu, J.I.: On Lagrange and Hermite interpolation in R n. Numer. Math. 39, 1–14 (1982)
Lee, S.L., Phillips, G.M.: Construction of lattices for Lagrange interpolation in projective space. Constr. Approx. 7, 283–297 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.
Rights and permissions
About this article
Cite this article
Carnicer, J.M., Godés, C. Classification of sets satisfying the geometric characterization. Numer Algor 50, 145–154 (2009). https://doi.org/10.1007/s11075-008-9221-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-008-9221-8