Approximation Orders of FSI Spaces in L 2 (R d )

. A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant subspace S(Φ) of L2(Rd) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators $\varphi\in\Phi$ of the subspace. Further, when the generators satisfy a certain technical condition, then, under the mild assumption that the set of 1-periodizations of the generators is linearly independent, such a space is shown to provide approximation order k if and only if ${\rm span}\{\varphi(\cdot-j): |j| < k, \varphi\in\Phi\}$ contains a ψ (necessarily unique) satisfying $D^j\hat{\psi}(\alpha)=\delta_j\delta_\alpha$ for $|j| < k$, $\alpha\in 2\pi{\Bbb Z}^d$ . The technical condition is satisfied, e.g., when the generators are $O(|\cdot|^{-\rho})$ at infinity for some ρ>k+d . In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2].