Approximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})

de Boor, Carl; DeVore, Ronald A.; Ron, Amos

doi:10.1090/S0002-9947-1994-1195508-X

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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