A characterization of the approximation order of translation invariant spaces of functions
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- by Rong Qing Jia PDF
- Proc. Amer. Math. Soc. 111 (1991), 61-70 Request permission
Abstract:
Let $S$ be a space of functions on $\mathbb {R}$ with the following properties: (i) $S$ is translation invariant, i.e., $f \in S$ implies $f( \cdot \pm 1) \in S$; (ii) $\operatorname {dim} S{|_{[0,1]}} < \infty$; (iii) $S$ is closed under uniform convergence on compact sets. In this paper we characterize the approximation order of $S$ by proving the following: Theorem. $S$ provides approximation of order $k$ if and only if $S$ contains a compactly supported function $\psi$ such that the Fourier transform $\hat \psi$ of $\psi$ satisfies $\hat \psi (0) = 1$ and ${D^\alpha }\hat \psi (2\pi j) = 0$ for $0 \leq \alpha < k$ and $j \in \mathbb {Z}\backslash \{ 0\}$. This result extends a corresponding result of de Boor and DeVore, who proved the above theorem for the case $k = 1$.References
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Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 61-70
- MSC: Primary 41A65
- DOI: https://doi.org/10.1090/S0002-9939-1991-1010801-1
- MathSciNet review: 1010801