Lp-error estimates for "shifted" surface spline interpolation on Sobolev space

Mathematics of Computation

Volumn 72, Issue 243, 2003, Pages 1349-1367

  Yoon, Jungho ^a  

^a Ewha Womans University   (South Korea)

Author keywords

"Shifted" surface spline; Interpolation; Radial basis function; Surface spline

Indexed keywords

EID: 0038129680     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-02-01498-9     Document Type: Article

References (27)

1. [AS] M. Abramowitz and I. Stegun, A Handbook of Mathematical Functions, Dover Publications, New York, 1972. MR 94b:00012 (reprint)
2. [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
3. [BrS] S.C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. MR 95f:65001
4. [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
5. [Bu2] M. D. Buhmann, Multivariate cardinal interpolation with radial basis functions, Constr. Approx. 6 (1990), 225-255. MR 91f:41001
6. [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
7. [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
8. [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
9. m-splines, RAIRO Analyse numerique 12 (1978), 325-334. MR 80j:41052
10. [F] G. B. Folland, Real Analysis, John Wiley & Sons, New York, 1984. MR 86k:28001
11. [GS] I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 1, Academic Press, 1964. MR 55:8786a
12. [H] W.W. Hager, Applied Numerical Linear Algebra, Englewood Cliffs, N.J., Prentice Hall, 1988.
13. [L] D. Levin, The approximation power of moving least-squares, Math. Comp. 67 (1998), 1517-1531. MR 99a:41039
14. [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
15. [M] C. A. Micchelli, Interpolation of Scattered Data: Distances, Matrices, and Conditionally Positive Functions, Constr. Approx. 2 (1986), 11-22. MR 88d:65016
16. [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
17. [MN2] W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive functions II, Math. Comp. 54 (1990), 211-230. MR 90e:41007
18. [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
19. [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
20. [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
21. [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
22. [S1] R. Schaback, Error Estimates and Condition Numbers for Radial Basis Function Interpolation, Adv. in Comp. Math. 3 (1995), 251-264. MR 96a:41004
23. [S2] R. Schaback, Approximation by Radial Basis Functions with Finitely Many Centers, Constr. Approx. 12 (1996), 331-340. MR 97d:41013
24. [S3] R. Schaback, Improved Error Bounds for Scattered Data Interpolation by Radial Basis Functions, Math. Comp. 68 (1999), 201-216. MR 99d:41037
25. [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
26. d) from a Space Spanned by the Scattered Shifts of a Radial Basis Function, Constr. Approx. 17 (2001), 227-247. MR 2002a:41021
27. [Y2] J. Yoon, Interpolation by Radial Basis Functions on Sobolev Space, J. of Approx. Th. 112 (2001), 1-15.