The uniqueness of the dual of Weyl–Heisenberg subspace frames

Abstract

From the Weyl–Heisenberg (WH) density theorem, it follows that a WH-frame (g mα,nβ) m,n∈ Z for L 2( R) has a unique WH-dual if and only if αβ=1. However, the same argument does not apply to the subspace WH-frame case and it is not clear how to use standard methods of Fourier analysis to deal with this situation. In this paper, we apply operator algebra theory to obtain a very simple necessary and sufficient condition for a given frame (induced by a projective unitary representation of a discrete group) to admit a unique dual (induced by the same system). As a special case, we obtain a characterization for the subspace WH-frames that have unique WH-duals (within the subspace). Using this characterization and the Zak transform, we are able to prove that if (g mα,nβ) m,n∈ Z is a WH-frame for a subspace M of L 2( R) , then, (i) (g mα,nβ) m,n∈ Z has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (g mα,nβ) m,n∈ Z has a unique WH-dual in M if and only if (g mα,nβ) m,n∈ Z is a Riesz sequence; (iii) if αβ<1, then the WH-dual for (g mα,nβ) m,n∈ Z in M is not unique.