## Approximation From Shift-Invariant Subspaces of ... (1991)

Venue: | Trans. Amer. Math. Soc |

Citations: | 130 - 38 self |

### BibTeX

@ARTICLE{Boor91approximationfrom,

author = {Carl de Boor and Ronald A. Devore and Amos Ron},

title = {Approximation From Shift-Invariant Subspaces of ...},

journal = {Trans. Amer. Math. Soc},

year = {1991},

volume = {341},

pages = {787--806}

}

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### Abstract

: A complete characterization is given of closed shift-invariant subspaces of L 2 (IR 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. AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30 Key Words and phrases: approximation order, Strang-Fix conditions, shift-invariant spaces, radial basis functions, orthogonal projection. Authors' affiliation and address: 1 Center for Mathematical Sciences University of Wisconsin-Madison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work...