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Shift-invariant spaces on the real line

Author: Rong-Qing Jia
Journal: Proc. Amer. Math. Soc. 125 (1997), 785-793
MSC (1991): Primary 41A25, 41A15, 46E30
MathSciNet review: 1350950
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Abstract: We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in $L_p(\mathbb {R})$ $(1\le p\le \infty )$. Based on a study of linear independence of the shifts of the generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on $\mathbb {Z}$. Moreover, we show that such a shift-invariant space provides $L_p$-approximation order $k$ if and only if it contains all polynomials of degree less than $k$.

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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2G1

Keywords: Shift-invariant spaces, approximation order
Received by editor(s): April 13, 1995
Received by editor(s) in revised form: August 10, 1995
Additional Notes: The author was supported in part by NSERC Canada under Grant OGP 121336
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society