Frame representations and Parseval duals with applications to Gabor frames

摘要

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 is an element of G} induced by a projective unitary representation pi of a group G, it is possible that {pi(g)xi : g is an element of 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 is an element of 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 G(g, L, 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 L x K is less than or equal to 1/2.