  We consider frames arising from the action of a unitary representation of a
discrete countable abelian group. We show that the range of the analysis
operator can be determined by computing which characters appear in the
representation. This allows one to compare the ranges of two such frames, which
is useful for determining similarity and also for multiplexing schemes. Our
results then partially extend to Bessel sequences arising from the action of
the group. We apply the results to sampling on bandlimited functions and to
wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two
sampling transforms to have orthogonal ranges, and two analysis operators for
wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient
condition is easy to compute in terms of the periodization of the Fourier
transform of the frame generators.
