Let E be a locally convex space
of temperate distributions on the n-dimensional Euclidean space Rn, and G a closed
subgroup of Gl(n,R), the general linear group over Rn. An attempt is made to
identify those distributions which can be approximated in E by linear combinations
of distributions of the form u(Ax + b), where u is a fixed element of E, A varies over
G, and b varies over Rn. A cancellation theorem is proved; this then allows the
support of the Fourier transform of any annihilator of the set of distributions of the
form u(Ax + b) to be localized. This in turn is used to obtain approximation
results.
