The local approximation order from a scale
(
S
h
)
({S_h})
of approximating functions on
R
m
{{\mathbf {R}}^m}
is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions
φ
\varphi
whose integer translates
φ
(
⋅
−
j
)
,
j
∈
z
m
\varphi ( \cdot - j),j \in {z^m}
, span the space
S
=
S
1
S = {S_1}
from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.