Frame representations and Parseval duals with applications to Gabor frames

Let x n \x₍\ be a frame for a Hilbert space H H. We investigate the conditions under which there exists a dual frame for x n \x₍\ which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether x n \x₍\ can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame π (g) ξ: g ∈ G \ (g): g G\ induced by a projective unitary representation π of a group G G, it is possible that π (g) ξ: g ∈ G \ (g): g G\ can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame π (g) ξ: g ∈ G