Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )

A complete characterization is given of closed shift-invariant subspaces of L2(Rd) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.