Recent developments in error estimates for scattered-data interpolation via radial basis functions

Numerical Algorithms

Volumn 39, Issue 1-3, 2005, Pages 307-315

  Narcowich, Francis J ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Band limited functions; Error estimates; Interpolation; Radial basis functions; Scattered data

Indexed keywords

EID: 17444404537     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11075-004-3644-7     Document Type: Article

References (25)

1. R.A. Brownlee and W. Light, Approximation orders for interpolation by surface splines to rough functions, IMA J. Numer. Anal., to appear.
2. M.D. Buhmann, Multivariate cardinal interpolation with radial basis functions, Constr. Approx. 6 (1990) 225-255.
3. M.D. Buhmann, New developments in the theory of radial basis function interpolation, in: Multivariate Approximation; From CAGD to Wavelets, Santiago (1992), eds. K. Jetter and F.I. Utreras (World Scientific, Singapore, 1993) pp. 35-75.
4. M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge Univ. Press, Cambridge, 2003).
5. d), Trans. Amer. Math. Soc. 341 (1994) 787-806.
6. J. Duchon, Splines minimizing rotation invariant semi-norms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables, Proc. of Conf., Math. Res. Inst., Oberwolfach (1976), Lecture Notes in Mathematics, Vol. 571 (Springer, Berlin, 1977) pp. 85-100.
7. m -splines, RAIRO Anal. Numér. 12(4) (1978) vi 325-334.
8. P. Erdos, On some convergence properties of the interpolation polynomials, Ann. Math. 44 (1943) 330-337.
9. M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, Vol. 79 (Amer. Math. Soc., Providence, RI, 1991).
10. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1977).
11. M. Johnson, A note on the limited stability of surface splines, Preprint (2002).
12. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988) 77-79.
13. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990) 211-230.
14. H.N. Mhaskar, F.J. Narcowich, N. Sivakumar and J.D. Ward, Approximation with interpolatory constraints, Proc. Amer. Math. Soc. 130 (2002) 1355-1364.
15. F.J. Narcowich and J.D. Ward, Scattered-data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal. 33 (2002) 1393-1410.
16. n: Error estimates for radial basis and band-limited functions, SIAM J. Math. Anal., to appear.
17. F.J. Narcowich, J.D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp., to appear.
18. R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996) 331-340.
19. R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp. 68 (1999) 201-216.
20. E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, NJ, 1971).
21. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, AICM 4 (1995) 389-396.
22. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258-272.
23. H. Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp. 68 (1999) 1521-1531.
24. Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13-27.
25. J. Yoon, Spectral approximation orders of radial basis function interpolation on the Sobolev space, SIAM J. Math. Anal. 33 (2001) 946-958.