$L_p$-error estimates for “shifted” surface spline interpolation on Sobolev space
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Abstract:
The accuracy of interpolation by a radial basis function $\phi$ is usually very satisfactory provided that the approximant $f$ is reasonably smooth. However, for functions which have smoothness below a certain order associated with the basis function $\phi$, no approximation power has yet been established. Hence, the purpose of this study is to discuss the $L_p$-approximation order ($1\leq p\leq \infty$) of interpolation to functions in the Sobolev space $W^k_p(\Omega )$ with $k> \max (0,d/2-d/p)$. We are particularly interested in using the “shifted” surface spline, which actually includes the cases of the multiquadric and the surface spline. Moreover, we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- B. J. C. Baxter, N. Sivakumar, and J. D. Ward, Regarding the $p$-norms of radial basis interpolation matrices, Constr. Approx. 10 (1994), no. 4, 451–468. MR 1288640, DOI 10.1007/BF01303522
- 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
- Kurt Jetter and Florencio I. Utreras (eds.), Multivariate approximation: from CAGD to wavelets, Series in Approximations and Decompositions, vol. 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1359541, DOI 10.1142/2066
- M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6 (1990), no. 3, 225–255. MR 1054754, DOI 10.1007/BF01890410
- N. Dyn, Interpolation and approximation by radial and related functions, Approximation theory VI, Vol. I (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 211–234. MR 1090994
- N. Dyn, I. R. H. Jackson, D. Levin, and A. Ron, On multivariate approximation by integer translates of a basis function, Israel J. Math. 78 (1992), no. 1, 95–130. MR 1194962, DOI 10.1007/BF02801574
- N. Dyn and A. Ron, Radial basis function approximation: from gridded centres to scattered centres, Proc. London Math. Soc. (3) 71 (1995), no. 1, 76–108. MR 1327934, DOI 10.1112/plms/s3-71.1.76
- 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
- Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- W.W. Hager, Applied Numerical Linear Algebra, Englewood Cliffs, N.J., Prentice Hall, 1988.
- David Levin, The approximation power of moving least-squares, Math. Comp. 67 (1998), no. 224, 1517–1531. MR 1474653, DOI 10.1090/S0025-5718-98-00974-0
- Will Light and Henry Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Theory 92 (1998), no. 2, 245–266. MR 1604931, DOI 10.1006/jath.1997.3118
- 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
- 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 S. A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. Approx. Theory 70 (1992), no. 1, 94–114. MR 1168377, DOI 10.1016/0021-9045(92)90058-V
- Francis J. Narcowich and Joseph D. Ward, Norms of inverses and condition numbers for matrices associated with scattered data, J. Approx. Theory 64 (1991), no. 1, 69–94. MR 1086096, DOI 10.1016/0021-9045(91)90087-Q
- Will Light (ed.), Advances in numerical analysis. Vol. II, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. Wavelets, subdivision algorithms, and radial basis functions. MR 1172118
- 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, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math. 3 (1995), no. 3, 251–264. MR 1325034, DOI 10.1007/BF02432002
- 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
- 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
- Zong Min Wu and Robert Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), no. 1, 13–27. MR 1199027, DOI 10.1093/imanum/13.1.13
- J. Yoon, Approximation in $L_p(\mathbf R^d)$ from a space spanned by the scattered shifts of a radial basis function, Constr. Approx. 17 (2001), no. 2, 227–247. MR 1814356, DOI 10.1007/s003650010033
- J. Yoon, Interpolation by Radial Basis Functions on Sobolev Space, J. of Approx. Th. 112 (2001), 1-15.
Additional Information
- Jungho Yoon
- Affiliation: Department of Mathematics, Ewha Women’s University, Dae Hyun-Dong, Seo Dae Moon-Gu, Seoul 120-750, Korea
- Email: yoon@math.ewha.ac.kr
- Received by editor(s): April 4, 2000
- Received by editor(s) in revised form: September 5, 2001
- Published electronically: December 18, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 1349-1367
- MSC (2000): Primary 41A05, 41A15, 41A25, 41A30, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-02-01498-9
- MathSciNet review: 1972740