Abstract
We present a survey of research on orbits of unitary representations developed under the guidance and leadership of Guido Weiss in the last 20 years. This line of research started via the study of wavelets and other reproducing function systems. It emphasizes the role of translations in such systems and leads naturally to the study of shift invariant spaces. The algebraic properties of integers influence many results in such spaces and suggest a more abstract nature of the entire approach. Unitary representations of various discrete groups provide far reaching generalizations of basic results. This topic is currently actively pursued by numerous authors, many of them being Guido’s collaborators for years.
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