In I. Daubechies' "Ten Lectures on Wavelets" it is stated in note 2 of chapter 2 (p. 51) that A. Grossmann, J. Morlet, and T. Paul's article "Transforms associated to square Integrable group representations. I. General results" can be generalized to cyclic representations. Unfortunately, Daubechies gives no proof of this generalization only citing private communication. Now, I have a really hard time seeing how to generalize this article, as the main ingredient of the article seems to be Schur's lemma, which is not applicable to cyclic representations. My question is therefore if someone knows of a proof of this alleged generalization?
As pointed out in the comments it is a bit unclear what it means to generalize a whole article, but this is actually Daubechies' claim in note 2. I can however be more specific, as my main question is wether or not theorem 3.1 in the article of Grossmann, Morlet, and Paul can be generalized to cyclic square integrable representations. This is the following result:
Theorem 3.1. Let $U$ be an irreducible square integrable representation of a locally compact Hausdorff group $G$, acting on the Hilbert space $\mathcal{H}$. Then there exists on $\mathcal{H}$ a unique unbounded self-adjoint positive operator $C$ such that the following hold.
(i) The set of admissible vectors coincides with the domain of $C$.
(ii) Let $g_1$ and $g_2$ be any two admissible vectors. Let $f_1$ and $f_2$ be any two vectors in $\mathcal{H}$. Then
$\displaystyle \int_G \langle f_1, U(x)g_1 \rangle \overline{\langle f_2, U(x)g_2 \rangle} \, dx = \langle Cg_2, Cg_1 \rangle \langle f_1, f_2 \rangle$.
By an admissible vector it is meant a vector $g \in \mathcal{H}$ such that $\langle g, U(x)g \rangle \in L^2(G)$.