Shift-invariant spaces on the real line

Jia, Rong-Qing

doi:10.1090/S0002-9939-97-03586-7

Shift-invariant spaces on the real line
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by Rong-Qing Jia

Proc. Amer. Math. Soc. 125 (1997), 785-793

DOI: https://doi.org/10.1090/S0002-9939-97-03586-7

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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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