Approximation orders for natural splines in arbitrary dimensions

Gutzmer, Tim; Melenk, Jens

doi:10.1090/S0025-5718-00-01299-0

Approximation orders for natural splines in arbitrary dimensions
HTML articles powered by AMS MathViewer

by Tim Gutzmer and Jens Markus Melenk PDF

Math. Comp. 70 (2001), 699-703 Request permission

Abstract:

Based on variational properties, we generalize the approximation properties of the univariate natural cubic spline to splines in arbitrary dimensions.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Marc Atteia, Fonctions “spline” et noyaux reproduisants d’Aronszajn-Bergman, Rev. Française Informat. Recherche Opérationnelle 4 (1970), no. R-3, 31–43 (French). MR 0300061
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Jean Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 10 (1976), no. R-3, 5–12 (French, with Loose English summary). MR 0470565
Jean Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^{m}$-splines, RAIRO Anal. Numér. 12 (1978), no. 4, 325–334, vi (French, with English summary). MR 519016, DOI 10.1051/m2an/1978120403251
Jean Meinguet, Multivariate interpolation at arbitrary points made simple, Z. Angew. Math. Phys. 30 (1979), no. 2, 292–304 (English, with French summary). MR 535987, DOI 10.1007/BF01601941
M. J. D. Powell, The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68 (1994), no. 1, 107–128. MR 1278451, DOI 10.1007/s002110050051
Robert Schaback, Radial basis functions viewed from cubic splines, Multivariate approximation and splines (Mannheim, 1996) Internat. Ser. Numer. Math., vol. 125, Birkhäuser, Basel, 1997, pp. 245–258. MR 1485009, DOI 10.1007/978-3-0348-8871-4_{2}0
R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp. 68 (1999), no. 225, 201–216. MR 1604379, DOI 10.1090/S0025-5718-99-01009-1