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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximation from locally finite-dimensional
shift-invariant spaces


Author: Kang Zhao
Journal: Proc. Amer. Math. Soc. 124 (1996), 1857-1867
MSC (1991): Primary 41A15, 41A25, 41A28, 41A63
MathSciNet review: 1307577
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Abstract: After exploring some topological properties of locally finite-dimensional shift-invariant subspaces $S$ of $L_p(\mathbb {R}^s)$, we show that if $S$ provides approximation order $k$, then it provides the corresponding simultaneous approximation order. In the case $S$ is generated by a compactly supported function in $L_\infty (\mathbb {R})$, it is proved that $S$ provides approximation order $k$ in the $L_p(\mathbb {R})$-norm with $p>1$ if and only if the generator is a derivative of a compactly supported function that satisfies the Strang-Fix conditions.


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Additional Information

Kang Zhao
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Address at time of publication: Structural Dynamics Research Corporation, 2000 Eastman Dr., Milford, Ohio 45150
Email: kang.zhao@sdrc.com

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03253-4
PII: S 0002-9939(96)03253-4
Keywords: Approximation order, locally finite-dimensional, polynomial reproducing, shift-invariant space, simultaneous approximation, Strang-Fix condition
Received by editor(s): June 28, 1994
Received by editor(s) in revised form: December 13, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society