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On radial basis approximation on periodic grids

Published online by Cambridge University Press:  24 October 2008

Martin D. Buhmann  and
Charles A. Micchelli
Martin D. Buhmann
Affiliation:
Magdalene College, University of Cambridge, Cambridge CB3 OAG
Charles A. Micchelli
Affiliation:
IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights NY 10598, U.S.A.
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Extract

A radial basis function approximation in n variables has the form

where ø:ℝn → ℝ denotes the n-variate, spherically symmetric function associated with a prescribed radial basis function ø+:ℝ+ → ℝ, i.e. ø = ø+(‖ · ‖), the norm being Euclidean. The are real coefficients (often, approximants s above are considered where only finitely many λjs are non-zero), and is a fixed set of points in ℝn (of course, only the xj with non-zero coefficient λj affect s). Thus s is a linear combination of translates of a radially symmetric function which can be of global support, the simplest choice being , where c is a positive parameter. The latter is referred to as the multiquadric function and is usefull in applications.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 317 - 334
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Beatson, R. K. and Powell, M. J. D.. Univariate multiquadric approximation: quasi- interpolation to scattered data. Report DAMTP 1990/NA7, University of Cambridge.Google Scholar
[2]Buhmann, M. B.. Multivariable interpolation using radial basis functions. Ph.D. thesis, University of Cambridge (1989).Google Scholar
[3]Buhmann, M. D. and Micchelli, C. A.. Multiply monotone functions for cardinal interpolation. Adv. in Appl. Math. 12 (1991), 358–386.CrossRefGoogle Scholar
[4]Buhmann, M. P. and Dyn, N.. Error estimates for multiquadric interpolation. In Curves and Surfaces (editors Laurent, P. J., LeMéhauté, A., Schumaker, L. L.). (Academic Press, 1991), pp. 51–58.CrossRefGoogle Scholar
[5]Jackson, I. R. H.. Radial basis function methods for multivariable approximation. Ph.D. thesis, University of Cambridge (1988).Google Scholar
[6]Jones, D. S.. The Theory of Generalised Functions (Cambridge University Press, 1982).CrossRefGoogle Scholar
[7]Karlin, S.. Total Positivity, vol. 1 (Stanford University Press, 1968).Google Scholar
[8]Light, W. A. and Cheney, E. W.. Quasi-interpolation with translates of a function having non-compact support. Constr. Approx. 8 (1992), 35–48.CrossRefGoogle Scholar
[9]Madych, W. R. and Nelson, S. A.. Multivariate interpolation and conditionally positive definite functions II. Math. Comp. 54 (1990), 211–230.CrossRefGoogle Scholar
[10]Powell, M. J. D.. The theory of radial basis function approximation in 1990. In Numerical Analysis II: Wavelets, Subdivision and Radial Basis Functions (editor Light, W. A.) (Oxford University Press, 1992), pp. 105–210.Google Scholar
[11]Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971).Google Scholar
[12]Stoer, J.. Einführung in die Numerische Mathematik, vol. 1 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[13]Wu, Z. and Schaback, R.. Local error estimates for radial basis function interpolation of scattered data. Report, University of Gottingen (1990).Google Scholar