The uniqueness of the dual of Weyl-Heisenberg subspace frames
Abbreviated Journal Title
Appl. Comput. Harmon. Anal.
Weyl-Heisenberg (Gabor) frame; Dual frame; Zak transform; projective; unitary representations; group-like unitary systems; von Neumann; algebras; Mathematics, Applied; Physics, Mathematical
From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (g(malpha,nbeta))(m,n is an element of Z) for L-2(R) has a unique WH-dual if and only if alphabeta = 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(malpha,nbeta))(m,nis an element ofZ) is a W-H-frame for a subspace M of L-2(R), then, (i) (g(malpha,nbeta))(m,nis an element ofZ) has a unique WH-dual in M when alphabeta is an integer; (ii) if alphabeta is irrational, then (g(malpha,nbeta))(m,nis an element ofZ) has a unique WH-dual in M if and only if (g(malpha,nbeta))(m,nis an element ofZ) is a Riesz sequence; (iii) if alphabeta < 1, then the WH-dual for (g(malpha,nbeta))(m,nis an element ofZ) in M is not unique. (C) 2004 Elsevier Inc. All rights reserved.
Applied and Computational Harmonic Analysis
"The uniqueness of the dual of Weyl-Heisenberg subspace frames" (2004). Faculty Bibliography 2000s. 4356.