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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.

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

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