Approximation orders of FSI spaces in L2(Rd)

A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant subspace S(Phi) of L-2(R-d) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators phi is an element of 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 span{phi(.-j) :\j\ < k, phi is an element of Phi} contains a psi (necessarily unique) satisfying D-j<(psi)over cap>(alpha) = delta(j)delta(alpha) for \j\ < k, alpha is an element of 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 result of Jia [J1], [J2].