Abstract
We investigate the approximation by manifolds ℋ n (φ) generated by linear combinations of n radial basis functions on Rd of the form φ(|⋅−a|), where φ is the thin-plate spline type function. We obtain exact asymptotic estimates for the approximation of Sobolev classes W r∞ (Bd) in the space L∞(Bd) on the unit ball Bd.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975).
A. Bejancu, The uniform convergence of multivariate natural splines, DAMTP Technical Report, University of Cambridge (1997).
A. Bejancu, On the accuracy of surface spline approximation and interpolation to bump functions, DAMTP Technical Report, University of Cambridge (2000).
M.D. Buhmann, Approximation and interpolation with radial functions, in: Multivariate Approximation and Applications, eds. N. Dyn (Cambridge Univ. Press, Cambridge, 2001) pp. 25–43.
M.D. Buhmann, N. Dyn and D. Levin, On quasi-interpolation by radial basis functions with scattered centres, Constr. Approx. 11 (1995) 239–254.
N. Dyn, I.R.H. Jackson, D. Levin and A. Ron, On multivariate approximation by integer translates of a basis function, Israel J. Math. 78, 95–130.
J. Duchon, Sur l’erreur d’interpolation des fonctions de plusiers variables par les Dm-splines, RAIRO Anal. Numér. 12 (1978) 325–334.
M.J. Johnson, A bound on the approximation order of surface splines, Constr. Approx. 14 (1998) 429–438.
M.J. Johnson, On the error in surface spline interpolation of a compactly supported function, manuscript, University of Kuwait (1998).
V. Maiorov, On best approximation by ridge functions, J. Approx. Theory 99 (1999) 68–94.
V. Maiorov, On best approximation by radial functions, J. Approx. Theory 120 (2003) 36–70.
V. Maiorov and R. Meir, Lower bounds for multivariate approximation by affine-invariant dictionaries, IEEE Trans. Inform. Theory 47 (2001) 1569–1575.
V. Maiorov and A. Pinkus, Lower bounds for approximation by MLP neural networks, Neurocomputing 25 (1999) 81–91.
M.J.D. Powell, The theory of radial basis approximation in: Advances in Numerical Analysis, Vol. 2, ed. W.A. Light (Oxford Univ. Press, Oxford, 1990) pp. 105–210.
A. Pinkus, n-Widths in Approximation Theory (Springer, Berlin, 1985).
R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996) 331–340.
R. Schaback, Error estimates and conditions numbers for radial basis functions interpolations, Adv. Comput. Math. 3 (1995) 251–264.
R. Schaback and H. Wendland, Characterization and construction of radial basis functions, in: Multivariate Approximation and Applications, eds. N. Dyn (Cambridge Univ. Press, Cambridge, 2001) pp. 1–24.
H. Wendland, Optimal approximation orders on L p for radial basis functions, East J. Approx. 6 (2000) 87–102.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by R. Schaback
AMS subject classification
41A25, 41A63, 65D07, 41A15
Rights and permissions
About this article
Cite this article
Maiorov, V. On lower bounds in radial basis approximation. Adv Comput Math 22, 103–113 (2005). https://doi.org/10.1007/s10444-004-1090-7
Issue Date:
DOI: https://doi.org/10.1007/s10444-004-1090-7