Square integrability of group representations on homogeneous spaces. I. Reproducing triples and frames

A connection between a class of positive operator valued measures on a Hilbert space and certain reproducing kernel Hilbert spaces leads to the concept of a reproducing triple. Any such object generates an overcomplete family of vectors, which has most of the attributes of the familiar coherent states. A particular case of such a triple is the notion of frame, which, in a discrete situation, coincides with the structure underlying nonorthogonal expansions. The abstract machinery developed here will be used in a second paper to give a general definition of a square integrable representation of a group and the associated coherent states.