Approximation orders of FSI spaces in L 2 ( IRd )

A second look at the authors’ ([BDR1], [BDR2]) characterization of the approximation order of a Finitely generated Shift-Invariant subspace S(Φ) of L2(IR) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators φ ∈ Φ 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{φ(· − j) : |j| < k,φ ∈ Φ} contains a ψ (necessarily unique) satisfying Dψ̂(α) = δjδα for |j| < k, α ∈ 2πZZ. The technical condition is satisfied, e.g., when the generators are O(| · |−ρ) at infinity for some ρ > k+ d. In the case of compactly supported generators, this recovers an earlier result of Jia ([J1], [J2]). AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30