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$L_p$-error estimates for ``shifted'' surface spline interpolation on Sobolev space

Author: Jungho Yoon
Journal: Math. Comp. 72 (2003), 1349-1367
MSC (2000): Primary 41A05, 41A15, 41A25, 41A30, 41A63
Published electronically: December 18, 2002
MathSciNet review: 1972740
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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.

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  • [AS] M. Abramowitz and I. Stegun, A Handbook of Mathematical Functions, Dover Publications, New York, 1972. MR 94b:00012 (reprint)
  • [BSW] B. J. C. Baxter, N. Sivakumar, and J. D. Ward, Regarding the p-Norms of Radial Basis Interpolation Matrices, Constr. Approx. 10 (1994), 451-468. MR 95e:41002
  • [BrS] S.C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods , Springer-Verlag, New York, 1994. MR 95f:65001
  • [Bu1] M. D. Buhmann, New Developments in the Theory of Radial Basis Function Interpolation, Multivariate Approximation: From CAGD to Wavelets (K. Jetter, F.I. Utreras eds.), World Scientific, Singapore, (1993), 35-75. MR 96f:65004
  • [Bu2] M. D. Buhmann, Multivariate cardinal interpolation with radial basis functions, Constr. Approx. 6 (1990), 225-255. MR 91f:41001
  • [D] N. Dyn, Interpolation and Approximation by Radial and Related Functions, Approximation Theory VI, (C. K. Chui, L. L. Schumaker and J. Ward eds.), Academic press, (1989), 211-234. MR 92d:41002
  • [DJLR] N. Dyn, I.R.H. Jackson, D. Levin, and A. Ron, On Multivariate Approximation by Integer Translates of a Basis Function, Israel Journal of Mathematics 78 (1992), 95-130. MR 94i:41024
  • [DR] N. Dyn and A. Ron, Radial basis function approximation: from gridded centers to scattered centers, Proc. London Math. Soc. 71 (1995), 76-108. MR 96f:41040
  • [Du] J. Duchon, Sur l'erreur d' interpolation des fonctions de plusieurs variables par les $D^m$-splines, RAIRO Analyse numerique 12 (1978), 325-334. MR 80j:41052
  • [F] G. B. Folland, Real Analysis, John Wiley & Sons, New York, 1984. MR 86k:28001
  • [GS] I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 1, Academic Press, 1964. MR 55:8786a
  • [H] W.W. Hager, Applied Numerical Linear Algebra, Englewood Cliffs, N.J., Prentice Hall, 1988.
  • [L] D. Levin, The approximation power of moving least-squares, Math. Comp. 67 (1998), 1517-1531. MR 99a:41039
  • [LW] W. Light and H. Wayne, On power functions and error estimats for radial basis function interpolation, J. of Approx. Th. 92 (1998), 245-266. MR 98m:41006
  • [M] C. A. Micchelli, Interpolation of Scattered Data: Distances, Matrices, and Conditionally Positive Functions, Constr. Approx. 2 (1986), 11-22. MR 88d:65016
  • [MN1] W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive functions I, Approximation Theory and its Applications 4 (1988), no 4, 77-89. MR 90e:41006
  • [MN2] W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive functions II, Math. Comp. 54 (1990), 211-230. MR 90e:41007
  • [MN3] W. R. Madych and S. A. Nelson, Bounds on Multivariate Polynomials and Exponential Error Estimates for Multiquadric Interpolation, J. of Approx. Th. 70 (1992), 94-114. MR 93f:41009
  • [NW] F. J. Narcowich and J. D. Ward, Norms of Inverses and Condition Numbers for Matrices Associated with Scattered Data, J. Approx. Theory 64 (1991), 69-94. MR 92b:65017
  • [P1] M. J. D. Powell, The Theory of Radial basis function approximation in 1990, Advances in Numerical Analysis Vol. II: Wavelets, Subdivision Algorithms and Radial Basis Functions (W.A. Light ed.), Oxford University Press, (1992), 105-210. MR 95c:41003
  • [P2] M. J. D. Powell, The uniform convergence of thin plate spline interpolation in two dimension, Numer. Math. 68 (1994), 107-128. MR 95c:41037
  • [S1] R. Schaback, Error Estimates and Condition Numbers for Radial Basis Function Interpolation , Adv. in Comp. Math. 3 (1995), 251-264. MR 96a:41004
  • [S2] R. Schaback, Approximation by Radial Basis Functions with Finitely Many Centers, Constr. Approx. 12 (1996), 331-340. MR 97d:41013
  • [S3] R. Schaback, Improved Error Bounds for Scattered Data Interpolation by Radial Basis Functions, Math. Comp. 68 (1999), 201-216. MR 99d:41037
  • [WS] Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27. MR 93m:65012
  • [Y1] J. Yoon, Approximation in $L^p(\mathbb R^d)$ from a Space Spanned by the Scattered Shifts of a Radial Basis Function, Constr. Approx. 17 (2001), 227-247. MR 2002a:41021
  • [Y2] J. Yoon, Interpolation by Radial Basis Functions on Sobolev Space, J. of Approx. Th. 112 (2001), 1-15.

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Additional Information

Jungho Yoon
Affiliation: Department of Mathematics, Ewha Women’s University, Dae Hyun-Dong, Seo Dae Moon-Gu, Seoul 120-750, Korea

Keywords: Radial basis function, interpolation, surface spline, ``shifted'' surface spline
Received by editor(s): April 4, 2000
Received by editor(s) in revised form: September 5, 2001
Published electronically: December 18, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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