Controlled Approximation and a Characterization of the Local Approximation Order

The local approximation order from a scale of approximating functions on <!-- MATH ${{\mathbf{R}}^m}$ --> is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions whose integer translates <!-- MATH $\varphi ( \cdot - j),j \in {z^m}$ --> , span the space from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.