Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups. (English) Zbl 1326.81099
Summary: In this paper we introduce and investigate the concept of reproducing 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 Gabor systems and state sufficient 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). 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.
MSC:
| 81R30 | Coherent states |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 22B05 | General properties and structure of LCA groups |
| 43A32 | Other transforms and operators of Fourier type |
| 42C15 | General harmonic expansions, frames |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
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