Let ℋ he a closed subgroup of
the group of linear transformations of Rn onto itself. Let hx denote the image of the
point x under the transformation h, and let 𝒢 be the transpose grotip of ℋ: i.e. its
elements are associated with matrices which are the transposes of those in ℋ.
For f in L2(Rn), let Cl{f;ℋ× Rn} denote the closure in the L2 norm of
the linear span of functions of the form f(hx + t) where h is in ℋ, and t
is in Rn. Since this space is translation-invariant, it is of the form L2(S):
i.e. the set of L2 functions r(x) such that the nonzero set of r, the Fourier
transform of r, is, except for a set of measure zero, included in S. In the first
theorem a precise description of S is given, and in the second, a function
is constmcted in a natural way whose translates alone generate the given
space.
