Abstract
Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Biermann, O.: Über näherungsweise Kubaturen. Monatsh. Math. Phys. 14, 211–225 (1903)
Biermann, O.: Vorlesungen über mathematische Näherungsmethoden. F. Vieweg, Braunschweig 1905.
Carnicer, J.M., Gasca, M.: Interpolation on lattices generated by cubic pencils, to appear in Adv. in Comp. Math.
Carnicer, J.M., Gasca, M.: Generation of lattices of points for bivariate interpolation. Numer. Algor. 39, 69–79 (2005)
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)
Gasca, M., Sauer, T.: Polynomial interpolation in several variables, Adv. Comp. Math. 12, 377–410 (2000)
Gasca, M., Sauer, T.: On the history of multivariate polynomial interpolation. JCAM, 122, 23–35 (2000)
Lee, S.L., Phillips, G.M.: Construction of Lattices for Lagrange Interpolation in Projective Space, Constr. Approx. 7, 283–297 (1991)
Lorentz, G.G.: Approximation of functions. Chelsea Publishing Company, New York, 1966
Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer, New York, 2003
Sauer, T., Yuan Xu : On multivariate Lagrange interpolation. Math. Comp. 64, 1147–1170 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.
Rights and permissions
About this article
Cite this article
Carnicer, J., Gasca, M. & Sauer, T. Interpolation lattices in several variables. Numer. Math. 102, 559–581 (2006). https://doi.org/10.1007/s00211-005-0667-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0667-5