Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting
HTML articles powered by AMS MathViewer
- by Francis J. Narcowich, Joseph D. Ward and Holger Wendland;
- Math. Comp. 74 (2005), 743-763
- DOI: https://doi.org/10.1090/S0025-5718-04-01708-9
- Published electronically: August 20, 2004
- PDF | Request permission
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.References
- 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
- R. Brownlee and W. Light, Approximation orders for interpolation by surface splines to rough functions, IMA J. Numer. Anal., 24 (2004), 179–192.
- Carl de Boor, Ronald A. DeVore, and Amos Ron, Approximation from shift-invariant subspaces of $L_2(\mathbf R^d)$, Trans. Amer. Math. Soc. 341 (1994), no. 2, 787–806. MR 1195508, DOI 10.1090/S0002-9947-1994-1195508-X
- Ronald A. DeVore and Robert C. Sharpley, Besov spaces on domains in $\textbf {R}^d$, Trans. Amer. Math. Soc. 335 (1993), no. 2, 843–864. MR 1152321, DOI 10.1090/S0002-9947-1993-1152321-6
- 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
- W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988), no. 4, 77–89. MR 986343
- W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions. II, Math. Comp. 54 (1990), no. 189, 211–230. MR 993931, DOI 10.1090/S0025-5718-1990-0993931-7
- W. R. Madych and E. H. Potter, An estimate for multivariate interpolation, J. Approx. Theory 43 (1985), no. 2, 132–139. MR 775781, DOI 10.1016/0021-9045(85)90121-2
- Charles A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), no. 1, 11–22. MR 891767, DOI 10.1007/BF01893414
- 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.
- F. J. Narcowich, J. D. Ward, and H. Wendland, Refined error estimates for radial basis function interpolation, Constr. Approx., 19 (2003), 541–564.
- Amos Ron, The $L_2$-approximation orders of principal shift-invariant spaces generated by a radial basis function, Numerical methods in approximation theory, Vol. 9 (Oberwolfach, 1991) Internat. Ser. Numer. Math., vol. 105, Birkhäuser, Basel, 1992, pp. 245–268. MR 1269365, DOI 10.1007/978-3-0348-8619-2_{1}4
- R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996), no. 3, 331–340. MR 1405002, DOI 10.1007/s003659900017
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 290095
- Holger Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), no. 4, 389–396. MR 1366510, DOI 10.1007/BF02123482
- Holger Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp. 68 (1999), no. 228, 1521–1531. MR 1648419, DOI 10.1090/S0025-5718-99-01102-3
- Holger Wendland, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal. 21 (2001), no. 1, 285–300. MR 1812276, DOI 10.1093/imanum/21.1.285
Bibliographic Information
- Francis J. Narcowich
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 129435
- Email: fnarc@math.tamu.edu
- Joseph D. Ward
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 180590
- Email: jward@math.tamu.edu
- Holger Wendland
- Affiliation: Universität Göttingen, Lotzestrasse 16-18, D-37083, Göttingen, Germany
- MR Author ID: 602098
- Email: wendland@math.uni-goettingen.de
- Received by editor(s): July 21, 2003
- Published electronically: 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 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 743-763
- MSC (2000): Primary 41A25; Secondary 41A05, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-04-01708-9
- MathSciNet review: 2114646