The uniqueness of the dual of Weyl-Heisenberg subspace frames

By Jean-Pierre Gabardo, Deguang Han. 2004

59 citations recorded by the source corpus.

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 (g mα,nβ ) m,n∈ Z in M is not unique.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/3ce410888b830ee47a75c1d2bacf8230da9e0b3f

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In Applied Mathematics

Citations in the corpus

Generalized Shift-Invariant Systems and Frames for Subspaces (2005, background): “Thus, it has many duals, but Gabardo and Han prove that only the canonical dual has the Gabor structure.”
Frame representations and Parseval duals with applications to Gabor frames (2008, background): “We need the following characterization result for uniqueness of the dual of the same type [18]:”