LOCAL APPROXIMATION ORDER OF BOX SPLINES

Let Δ denote the mesh on the plane R={（x, x）: x, x∈R} obtained by multiinteger translates of the three lines x=0, x=0 and x-x=0. A function s defined on Ris called a spline of degree k on the mesh Δ, if s agrees with a polynomial of （total） degree≤k on each component of R\Δ. Let π, Δ be the space of all spline functions of degree k on Δ, and π, Δ:=π, Δ∩C. Denote by Φ, p the set of all box splines in π, Δ, and by m（k, ρ） the local approximation order of Φ, ρ. So far, only few results about m（k, ρ） have been obtained by de Boor and Hllig, and Dahmen and Micchelli. In this paper, it is demonstratedthat the local approximation order m（k, ρ） is （1） 2k-2ρ, if 2k-3ρ =2, （2） 2k-2ρ-1, if 2k-3ρ=3 or 4, （3） k+1, if ρ=0, （4） min {2k-2ρ-2, k}, if 2k-3ρ≥5 and ρ≥1.Thus the problem of the local approximation order of box splines is solved completely.