Dual Integrable Representations on Locally Compact Groups

Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its introduction, the notion was extended to some specific classes of non-abelian groups, and a natural problem emerged, i.e., whether it can be extended to the entire class of (including non-abelian) locally compact groups. In this paper we solve this problem. Interestingly enough, the bracket function (or rather an operator in this case) keeps all of its useful properties, except the Cauchy-Schwarz inequality. Nevertheless, we show that cyclic spaces still can be represented in terms of bracket-weighted spaces.