APPROXIMATION ORDERS OF FSI SPACES IN L-2(R-D)

A second look at the authors' [BDR1], [BDR2] characterization of the a pproximation order of a Finitely generated Shift-Invariant (FSI) subsp ace S(Phi) of L-2(R-d) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators phi epsilon Phi , of the subspace. Further, when the generators satisfy a certain tech nical condition, then, under the mild assumption that the set of 1-per iodizations of the generators is linearly independent, such a space is shown to provide approximation order k if and only if span{phi(.-j) : \j\ < k, phi epsilon Phi} contains a psi (necessarily unique) satisfyi ng D-j <(psi)over cap> (alpha) = delta(j)delta(alpha) for \j\ < k, alp ha epsilon 2 pi Z(d). The technical condition is satisfied, e.g., when the generators are O(\.\(-rho)) at infinity for some rho > k + d. In the case of compactly supported generators, this recovers an earlier r esult of Jia [J1], [J2].