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Inverse and saturation theorems for radial basis function interpolation


Authors: Robert Schaback and Holger Wendland
Journal: Math. Comp. 71 (2002), 669-681
MSC (2000): Primary 41A05, 41A17, 41A27, 41A30, 41A40
DOI: https://doi.org/10.1090/S0025-5718-01-01383-7
Published electronically: November 28, 2001
MathSciNet review: 1885620
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Abstract: While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline interpolation we also give certain saturation theorems.


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  • 1. Bejancu, A., Local accuracy for radial basis function interpolation on finite uniform grids, J. of Approx. Theory 99 (1999) 242-257. MR 2000e:41002
  • 2. DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Springer, Berlin, 1993. MR 95f:41001
  • 3. Duchon, J., Sur l'erreur d'interpolation des fonctions de plusiers variables par les $D^m$-splines, R.A.I.R.O. Analyse numérique 12 (1978), 325-334. MR 80j:41052
  • 4. Golitschek, M. von, and W. Light, Interpolation by polynomials and radial basis functions on spheres, Constructive Approximation 17 (2000), 1-18. CMP 2001:04
  • 5. Jetter, K., J. Stoeckler, and J.D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp. 68 (1999), 733-747. MR 99i:41032
  • 6. John, F., Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience Publishers, Inc., New York, 1955. MR 17:746d
  • 7. Light, W. and Wayne, H., On power functions and error estimates for radial basis function interpolation, J. of Approx. Theory 92 (1998), 245-266. MR 98m:41006
  • 8. Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive definite functions, Approx. Theory and its Appl. 4 (1988), 77-89 MR 90e:41006
  • 9. Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990), 211-230. MR 90e:41007
  • 10. Schumaker, L. L., Spline Functions: Basic Theory, John Wiley & Sons, New York,1981. MR 82j:41001
  • 11. Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Advances in Comp. Math. 3 (1995), 251-264. MR 96a:41004
  • 12. Schaback, R., Improved error bounds for radial basis function interpolation, Math. Comp. 68 (1999), 201-216. MR 99d:41037
  • 13. Schaback, R., Native spaces for radial basis functions I, in: M.W. Müller, M.D. Buhmann, D.H. Mache, and M.Felten (eds.), New Developments in Approximation Theory, ISNM Vol. 132, 255-282, Birkhäuser, Basel, 1999. CMP 2000:05
  • 14. Schaback, R., A unified theory of radial basis functions (Native Hilbert spaces for radial basis functions II), J. of Computational and Applied Mathematics 121 (2000) 165-177. CMP 2001:01
  • 15. Wendland, H., Sobolev-type error estimates for interpolation by radial basis functions, in: A. LeMéhauté, C. Rabut, and L.L Schumaker (eds.), Surface Fitting and Multiresolution Methods, 337-344, Vanderbilt University Press, Nashville, TN, 1997. MR 99j:65012
  • 16. Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998), 258-272. MR 99g:65015
  • 17. Wu, Z. and Schaback, R. Local error estimates for radial basis function interpolation of scattered data, IMA J. of Numerical Analysis 13 (1993), 13-27. MR 93m:65012

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Additional Information

Robert Schaback
Affiliation: Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
Email: schaback@math.uni-goettingen.de

Holger Wendland
Affiliation: Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
Email: wendland@math.uni-goettingen.de

DOI: https://doi.org/10.1090/S0025-5718-01-01383-7
Keywords: Positive definite functions, approximation orders
Received by editor(s): February 10, 2000
Published electronically: November 28, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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