# Reproducing subgroups of affine Weyl-Heisenberg group

In this thesis we study a class of subgroups of the affine Weyl-Heisenberg group $G_{aWH}$of the type $\mathbb{T}\times \mathbb{R}^n\times V \rtimes H$, where $V$ is a non-trivial subspace of Euclidean space and $H$ is a non-trivial subgroup of the general linear group. The main purpose of this is to characterize reproducing subgroups of $G_{aWH}$ of the mentioned form. This class of subgroups contains standard wavelet transforms as well as windowed wavelet transforms, of the Weyl-Heisenberg group. We consider the construction of the continuous wavelet transforms on these subgroups, with the unitary representation $\pi(x,\xi,h,z) = z T_x M_\xi D_h$. We then give sharp admissibility criteria for a pair $(V,H)$ that have an admissible vector. Furthermore, we study invariant subspaces by focusing on the dual action of $V\rtimes H$ on Euclidean space, to approach square-integrability. Finally, we provide a new class of examples of reproducing subgroups of the type $\mathbb{T}\times \mathbb{R}^n\times V \rtimes H$ in dimension two and three. We also sketch the general structure of admissible subgroups in higher dimension.