Approximation orders of shift-invariant subspaces of W^s_2({ Bbb R}^d)

We extend the existing theory of approximation orders provided by nshift-invariant subspaces of L_2 to the setting of Sobolev spaces, provide ntreatment of L_2 cases that have not been covered before, and apply our nresults to determine approximation order of solutions to a refinement equation nwith a higher-dimensional solution space.