Order of linear approximation from shift-invariant spaces

A Fourier analysis approach is taken to investigate the approximation order of scaled versions of certain linear operators into shift-invariant subspaces ofL ₂(R ^d ). Quasi-interpolants and cardinal interpolants are special operators of this type, and we give a complete characterization of the order in terms of some type of ellipticity condition for a related function. We apply these results by showing that theL ₂-approximation order of a closed shift-invariant subspace can often be realized by such an operator.