The theory of reproducing systems on locally compact abelian groups
Volume 106 / 2006
Colloquium Mathematicum 106 (2006), 197-220
MSC: 43A70, 43A40, 43A15.
DOI: 10.4064/cm106-2-3
Abstract
A reproducing system is a countable collection of functions $\{\phi_j: j \in {\cal J}\}$ such that a general function $f$ can be decomposed as $f = \sum_{j \in {\cal J}} c_j(f) \, \phi_j$, with some control on the analyzing coefficients $c_j(f)$. Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups $G$. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on $L^2(G)$. As an application, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on $L^2(G)$.
