Approximation from shift-invariant subspaces of $L_ 2(\mathbf {R}^ d)$
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
DOI:
https://doi.org/10.1090/S0002-9947-1994-1195508-X
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
Keywords:
Approximation order,
Strang-Fix conditions,
shift-invariant spaces,
radial basis functions,
orthogonal projection
Article copyright:
© Copyright 1994
American Mathematical Society