Approximation Orders of FSI Spaces in L2(Rd) Academic Article uri icon


  • A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant (FSI) subspace S(Φ) of L2(ℝd) 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 Dj ψ(α) = δjδα for |j| < k, α ∈ 2πℤd. The technical condition is satisfied, e.g., when the generators are O(| · |-p) at infinity for some ρ > k + d. In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2].

author list (cited authors)

  • de Boor, C., DeVore, R. A., & Ron, A.

citation count

  • 33

publication date

  • October 1998