Frame representations and Parseval duals with applications to Gabor frames

By Deguang Han. 2008

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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 {π(g)ξ: g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π(g)ξ: 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} (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 × K is less than or equal to 1/2.

Subfields: Applied Mathematics

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