Operator-valued frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on (Formula presented.), restricted to (Formula presented.), forms a Hilbert-space frame for (Formula presented.). For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group (Formula presented.), where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of (Formula presented.) for suitable subsets (Formula presented.) of (Formula presented.) in terms of representations of (Formula presented.).