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Discrete least squares approximation and prewavelets from radial function spaces

Published online by Cambridge University Press:  24 October 2008

M. D. Buhmann
M. D. Buhmann
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

In this article we study the convergence behaviour of least squares approximations of various types by radial basis functions, i.e. least squares approximations from spaces spanned by radially symmetric functions φ (‖· −xj‖). Here the xj are given ‘centres’ in ℝn which we assume to lie on a grid. The inner products with respect to which the least squares problem is considered are discrete and Sobolev, i.e. may involve derivative information. Favourable estimates for the least squares errors are found that are shown to decrease as powers of the gridspacing. The work on discrete least squares approximations also gives rise to the construction of prewavelets from radial function spaces with respect to a discrete Sobolev inner product, which are discussed in the paper as well.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 3 , November 1993 , pp. 533 - 558
Copyright
Copyright © Cambridge Philosophical Society 1993

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