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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A characterization of the approximation order of translation invariant spaces of functions

Author: Rong Qing Jia
Journal: Proc. Amer. Math. Soc. 111 (1991), 61-70
MSC: Primary 41A65
MathSciNet review: 1010801
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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{\vert _{[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$.

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Keywords: Order of approximation, translation invariance
Article copyright: © Copyright 1991 American Mathematical Society