## The $L_{2}$-approximation order of surface spline interpolation

HTML articles powered by AMS MathViewer

- by Michael J. Johnson PDF
- Math. Comp.
**70**(2001), 719-737 Request permission

## Abstract:

We show that if the open, bounded domain $\Omega \subset \mathbb {R}^{d}$ has a sufficiently smooth boundary and if the data function $f$ is sufficiently smooth, then the $L_{p}(\Omega )$-norm of the error between $f$ and its surface spline interpolant is $O(\delta ^{\gamma _{p}+1/2})$ ($1\leq p\leq \infty$), where $\gamma _{p}:=\min \{m,m-d/2+d/p\}$ and $m$ is an integer parameter specifying the surface spline. In case $p=2$, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the $L_{2}$-approximation order of surface spline interpolation is $m+1/2$.## References

- Robert A. Adams,
*Sobolev spaces*, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0450957** - Shmuel Agmon,
*Lectures on elliptic boundary value problems*, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR**0178246** - Aurelian Bejancu Jr.,
*Local accuracy for radial basis function interpolation on finite uniform grids*, J. Approx. Theory**99**(1999), no. 2, 242–257. MR**1702221**, DOI 10.1006/jath.1999.3332 - M. D. Buhmann,
*Multivariate cardinal interpolation with radial-basis functions*, Constr. Approx.**6**(1990), no. 3, 225–255. MR**1054754**, DOI 10.1007/BF01890410 - Buhmann, M.D.,
*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, pp. 35–75. - M. D. Buhmann, N. Dyn, and D. Levin,
*On quasi-interpolation by radial basis functions with scattered centres*, Constr. Approx.**11**(1995), no. 2, 239–254. MR**1342386**, DOI 10.1007/BF01203417 - Jean Duchon,
*Splines minimizing rotation-invariant semi-norms in Sobolev spaces*, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, pp. 85–100. MR**0493110** - 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 - 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 - 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** - Rong Qing Jia and Junjiang Lei,
*Approximation by multi-integer translates of functions having global support*, J. Approx. Theory**72**(1993), no. 1, 2–23. MR**1198369**, DOI 10.1006/jath.1993.1002 - M. J. Johnson,
*A bound on the approximation order of surface splines*, Constr. Approx.**14**(1998), no. 3, 429–438. MR**1626718**, DOI 10.1007/s003659900082 - Johnson, M.J.,
*An improved order of approximation for thin-plate spline interpolation in the unit disk*, Numer. Math.**84**(2000), 451–474. - Johnson, M.J.,
*On the error in surface spline interpolation of a compactly supported function*, manuscript. - Will Light and Henry Wayne,
*Spaces of distributions, interpolation by translates of a basis function and error estimates*, Numer. Math.**81**(1999), no. 3, 415–450. MR**1668091**, DOI 10.1007/s002110050398 - 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 - Jaak Peetre,
*New thoughts on Besov spaces*, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR**0461123** - Powell, M.J.D.,
*The theory of radial basis function approximation in 1990*, Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Functions (W.A. Light, ed.), Oxford University Press, Oxford, 1992, pp. 105–210. - 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 - Walter Rudin,
*Principles of mathematical analysis*, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR**0385023** - Walter Rudin,
*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157** - 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,
*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 - Hans Triebel,
*Theory of function spaces. II*, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR**1163193**, DOI 10.1007/978-3-0346-0419-2 - Hans Triebel,
*Fractals and spectra*, Monographs in Mathematics, vol. 91, Birkhäuser Verlag, Basel, 1997. Related to Fourier analysis and function spaces. MR**1484417**, DOI 10.1007/978-3-0348-0034-1 - 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

## Additional Information

**Michael J. Johnson**- Affiliation: Deptartment of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait
- Email: johnson@mcc.sci.kuniv.edu.kw
- Received by editor(s): June 10, 1999
- Published electronically: October 27, 2000
- Additional Notes: This work was supported by Kuwait University Research Grant SM-175.
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp.
**70**(2001), 719-737 - MSC (2000): Primary 41A15, 41A25, 41A63, 65D05
- DOI: https://doi.org/10.1090/S0025-5718-00-01301-6
- MathSciNet review: 1813145