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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Approximation from shift-invariant subspaces of $ L\sb 2(\bold R\sp d)$


Authors: Carl de Boor, Ronald A. DeVore and Amos Ron
Journal: Trans. Amer. Math. Soc. 341 (1994), 787-806
MSC: Primary 41A25; Secondary 41A63, 42B10, 46E20
MathSciNet review: 1195508
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Abstract: A complete characterization is given of closed shift-invariant subspaces of $ {L_2}({\mathbb{R}^d})$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1195508-X
PII: S 0002-9947(1994)1195508-X
Keywords: Approximation order, Strang-Fix conditions, shift-invariant spaces, radial basis functions, orthogonal projection
Article copyright: © Copyright 1994 American Mathematical Society