Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups

Abstract: In this paper we introduce and investigate the concept of repro- ducing pairs as a generalization of continuous frames. Reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages. Moreover, we will investigate certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as a continuous version of nonstationary Ga- bor systems and state sufficient conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we iden- tify the structure of the frame operator (resp. resolution operator). We will apply our results to systems generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$. This setup will also serve as a nontrivial example of a system for which, whereas continuous frames exist, no dual system with the same structure exists even if we drop the frame property.