Simultaneous Approximation and Quasi-Interpolants Original Research Article

It is shown that the shift-invariant spaceS(ϕ) generated byϕ∈Wm2(Rs) provides simultaneous approximation orderk, withk>m, iffS(Dαϕ) provides approximation orderk−min theL2(Rs)-norm to all functions inDαWk2(Rs) for each |α|=m. Without appealing to the argument of polynomial reproduction, an explicit formula is presented for construction of quasi-interpolant of semi-discrete convolution type that achieves the approximation order provided byS(ϕ). The traditional condition that the symbol ∑α∈Zs ϕ(α) eαdoes not vanish on [−π…π]sis reduced to that it does not vanish on a neighborhood of the origin.