Mathematics of Computation

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

Author(s): Francis J. Narcowich; Joseph D. Ward; Holger Wendland.
Journal: Math. Comp. 74 (2005), 743-763.
MSC (2000): Primary 41A25; Secondary 41A05, 41A63
Posted: August 20, 2004
Retrieve article in: PDF DVI PostScript

Abstract: In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

1.: S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer, New York, 1994. MR 95f:65001
2.: R. Brownlee and W. Light, Approximation orders for interpolation by surface splines to rough functions, IMA J. Numer. Anal., 24 (2004), 179-192.
3.: C. de Boor, R. A. DeVore, and A. Ron, Approximation from shift-invariant subspaces of $L_2(\mathbb{R} ^d)$ , Trans. Amer. Math. Soc., 341 (1994), 787-806. MR 94d:41028
4.: R. A. DeVore and R. C. Sharpley, Besov spaces on domains in $\mathbb{R} ^d$ , Trans. Amer. Math. Soc., 335 (1993), 843-864. MR 93d:46051
5.: J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les -splines, Rev. Française Automat. Informat. Rech. Opér. Anal. Numer., 12 (1978), 325-334. MR 80j:41052
6.: W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approximation Theory Appl., 4 (1988), 77-89. MR 90e:41006
7.: -, Multivariate interpolation and conditionally positive definite functions II, Math. Comp., 54 (1990), 211-230. MR 90e:41007
8.: W. R. Madych and E. H. Potter, An estimate for multivariate interpolation, J. Approx. Theory, 43 (1985), 132-139. MR 86g:65022
9.: C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11-22. MR 88d:65016
10.: F. J. Narcowich and J. D. Ward, Scattered-data interpolation on $\mathbb{R} ^n:$ Error estimates for radial basis and band-limited functions, SIAM J. Math. Anal., to appear.
11.: F. J. Narcowich, J. D. Ward, and H. Wendland, Refined error estimates for radial basis function interpolation, Constr. Approx., 19 (2003), 541-564.
12.: A. Ron, The -approximation orders of principal shift-invariant spaces generated by a radial basis function, Numerical Methods in Approximation Theory. Vol. 9: Proceedings of the conference held in Oberwolfach, Germany, November 24-30, 1991 (Basel) (D. Braess et al., eds.), Int. Ser. Numer. Math., vol. 105, Birkhäuser, 1992, pp. 245-268. MR 95d:41035
13.: R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx., 12 (1996), 331-340. MR 97d:41013
14.: E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1971. MR 44:7280
15.: H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comp. Math., 4 (1995), 389-396. MR 96h:41025
16.: -, Meshless Galerkin methods using radial basis functions, Math. Comp., 68 (1999), 1521-1531. MR 99m:65221
17.: -, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal., 21 (2001), 285-300. MR 2002a:65025

Retrieve articles in Mathematics of Computation with MSC (2000): 41A25, 41A05, 41A63

Francis J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: fnarc@math.tamu.edu

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: jward@math.tamu.edu

Holger Wendland
Affiliation: Universität Göttingen, Lotzestrasse 16-18, D-37083, Göttingen, Germany
Email: wendland@math.uni-goettingen.de

DOI: 10.1090/S0025-5718-04-01708-9
PII: S 0025-5718(04)01708-9
Keywords: Radial basis functions, Sobolev error estimates, scattered zeros, scattered data.
Received by editor(s): July 21, 2003
Posted: August 20, 2004
Additional Notes: Research of the first author was supported by grant DMS-0204449 from the National Science Foundation.
Research of the second author was supported by grants DMS-9971276 and DMS-0204449 from the National Science Foundation.
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS