  Starting with the canonical coherent states, we demonstrate that all the
so-called nonlinear coherent states, used in the physical literature, as well
as large classes of other generalized coherent states, can be obtained by
changes of bases in the underlying Hilbert space. This observation leads to an
interesting duality between pairs of generalized coherent states, bringing into
play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in
each dual pair of families of nonlinear coherent states, at least one family is
related to a (generally) non-unitary projective representation of the
Weyl-Heisenberg group, which can then be thought of as characterizing the dual
pair.
