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On the continuity of multivariate Lagrange interpolation at natural lattices

Part of: Numerical approximation and computational geometry Approximations and expansions

Published online by Cambridge University Press:  10 April 2013

J.-P. Calvi  and
V. M. Phung
J.-P. Calvi
Affiliation:
Institut de MathématiquesUniversité de Toulouse III and CNRS (UMR 5219)31062, Toulouse Cedex 9, France email [email protected]
V. M. Phung
Affiliation:
Department of MathematicsHanoi University of Education136 Xuan Thuy StreetCaugiay, Hanoi, Vietnam email [email protected]

Abstract

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We give a natural geometric condition that ensures that sequences of interpolation polynomials (of fixed degree) of sufficiently differentiable functions with respect to the natural lattices introduced by Chung and Yao converge to a Taylor polynomial.

MSC classification

Secondary: 41A05: Interpolation 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 41A80: Remainders in approximation formulas 65D05: Interpolation
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 45 - 60
Copyright
© The Author(s) 2013 

References

Bloom, T. and Calvi, J.-P., ‘A continuity property of multivariate Lagrange interpolation’, Math. Comp. 66 (1997) no. 220, 1561–1577.CrossRefGoogle Scholar
Carnicer, J., Gasca, M. and Sauer, T., ‘Aitken–Neville sets, principal lattices and divided differences’, J. Approx. Theory 156 (2009) no. 2, 154–172.CrossRefGoogle Scholar
Chung, K. C. and Yao, T. H., ‘On lattices admitting unique Lagrange interpolations’, SIAM J. Numer. Anal. 14 (1977) no. 4, 735–743.CrossRefGoogle Scholar
de Boor, C., ‘The error in polynomial tensor-product, and Chung-Yao interpolation’, Surface fitting and multiresolution methods Chamonix–Mont–Blanc, 1996 (Vanderbilt University Press, Nashville, TN, 1997) 35–50.Google Scholar
de Boor, C. and Shekhtman, B., ‘On the pointwise limits of bivariate Lagrange projectors’, Linear Algebra Appl. 429 (2008) no. 1, 311–325.CrossRefGoogle Scholar
Gasca, M. and Sauer, T., ‘On bivariate Hermite interpolation with minimal degree polynomials’, SIAM J. Numer. Anal. 37 (2000) no. 3, 772–798.CrossRefGoogle Scholar
Micchelli, C. A., ‘A constructive approach to Kergin interpolation in ${\mathbf{R} }^{k} $ : multivariate $B$ -splines and Lagrange interpolation’, Rocky Mountain J. Math. 10 (1980) no. 3, 485–497.CrossRefGoogle Scholar
Sauer, T., ‘Polynomial interpolation in several variables: lattices, differences, and ideals’, Topics in multivariate approximation and interpolation, Studies in Computational Mathematics 12 (Elsevier, Amsterdam, 2006) 191–230.CrossRefGoogle Scholar
Sauer, T. and Xu, Y., ‘Regular points for Lagrange interpolation on the unit disk’, Numer. Algorithms 12 (1996) no. 3–4, 287–296.CrossRefGoogle Scholar
Shekhtman, B., ‘On a conjecture of Carl de Boor regarding the limits of Lagrange interpolants’, Constr. Approx. 24 (2006) 365–370, doi:10.1007/s00365-006-0634-7.CrossRefGoogle Scholar
Shekhtman, B., ‘On the limits of Lagrange projectors’, Constr. Approx. 29 (2009) 293–301, doi:10.1007/s00365-008-9016-0.CrossRefGoogle Scholar