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

A second look at the authors' ( BDR1], BDR2]) characterization of the approximation order of a Finitely generated Shift-Invariant subspace S( ) of L 2 (IR d ) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators ' 2 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 spanf'( ? j) : jjj < k; ' 2 g contains a (necessarily unique) satisfying D j b ( ) = j for jjj < k, 2 2 ZZ d . The technical condition is satis ed, e.g., when the generators are O(j j ? ) at in nity for some > k + d. In the case of compactly supported generators, this recovers an earlier result of Jia ( J1], J2]).