Frame representations and Parseval duals with applications to Gabor frames

Abstract:

Let $\{x_{n}\}$ be a frame for a Hilbert space $H$. We investigate the conditions under which there exists a dual frame for $\{x_{n}\}$ which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether $\{x_{n}\}$ 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 $\{\pi (g)\xi : g\in G\}$ induced by a projective unitary representation $\pi$ of a group $G$, it is possible that $\{\pi (g)\xi : g\in G\}$ 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 $\pi$ such that every frame $\{\pi (g)\xi : g\in G\}$ (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame $\textbf {G}(g, \mathcal {L}, \mathcal {K})$ (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of $\mathcal {L}\times \mathcal {K}$ is less than or equal to $\frac {1}{2}$.