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Approximation Orders of FSI Spaces in L2(R d)

By C. De Boor, R. A. Devore and A. Ron


A second look at the authors’ [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant (FSI) subspace S(�) of L2(Rd) results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators 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 ∈ �} contains a ψ (necessarily unique) satisfying D jψ(α) = δj δα for | j | < k, α ∈ 2πZd. 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]

Year: 1998
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