The theory of reproducing systems on locally compact abelian groups

A reproducing system is a countable collection of functions fj : j 2 Jg such that a general function f can be decomposed as f = P j2J cj(f)`j, with some control on the analyzing coe-cients cj(f). Several such systems have been introduced very successfully in mathematics and its applications. We present a unifled viewpoint to 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 of this result, we obtain a new characterization of Parseval frame generators for Gabor and a-ne systems on L 2 (G).

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