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

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 $k = 1$.