Measures and operators associated with Parseval distribution frames

Continuing the study by Tschinke et al. (2019), we examine further aspects of distribution frames (namely, Gel’fand and Parseval), particularly regarding those that are more relevant for applications in quantum physics. Parseval distribution frames are, in particular, closely related to coherent states. Thus, POV measures, Naimark dilations, and operators defined by Parseval distribution frames are the main subjects of this paper. The main results are Theorems 2.2 and 3.1. Theorem 2.2 gives a sufficient conditions for the existence of such distribution coherent states for positive operator valued measures. Theorem 3.1 establishes conditions under which the distribution coherent states can be identified with the projections of some Gel’fand distribution basis in a larger Hilbert space (in Naimark's sense).