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

Abstract.

A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant (FSI) subspace $S(\Phi)$ of L ₂(R ^d) 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 $\mathop{\rm span}\nolimits\{\varphi(\cdot-j): |j| < k, \varphi\in\Phi\}$ contains a $\psi$ (necessarily unique) satisfying $D^j\widehat{\psi}(\alpha)=\delta_j\delta_\alpha$ 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 $\rho$ >k+d. In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2].