The continuous nonstationary Gabor transform on LCA groups with applications to representations of the affine Weyl-Heisenberg group

Abstract: In this paper we introduce and investigate the concept of reproducing pairs which generalizes continuous frames. We will introduce a concept that represents a unifying way to look at certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as continuous nonstationary Gabor systems and investigate conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we identify the structure of the frame operator (resp. resolution operator). Moreover, we ask the question, whether there always exist mutually dual systems with the same structure such that the resolution operator is given by the identity, i.e. given $A:X\rightarrow B(\mathcal{H})$, if there exist $\psi,\varphi\in\mathcal{H}$, s.t. \begin{equation*} f=\int_X \langle f,A(x)\psi\rangle A(x)\varphi d\mu(x),\ \ \forall f\in \mathcal{H} \end{equation*} and show that the answer is not affirmative. As a counterexample we use a system generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$.