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- W2024762093 abstract "From the Weyl–Heisenberg (WH) density theorem, it follows that a WH-frame (gmα,nβ)m,n∈Z for L2(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 (gmα,nβ)m,n∈Z is a WH-frame for a subspace M of L2(R), then, (i) (gmα,nβ)m,n∈Z has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (gmα,nβ)m,n∈Z has a unique WH-dual in M if and only if (gmα,nβ)m,n∈Z is a Riesz sequence; (iii) if αβ<1, then the WH-dual for (gmα,nβ)m,n∈Z in M is not unique." @default.
- W2024762093 created "2016-06-24" @default.
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- W2024762093 date "2004-09-01" @default.
- W2024762093 modified "2023-10-16" @default.
- W2024762093 title "The uniqueness of the dual of Weyl–Heisenberg subspace frames" @default.
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- W2024762093 doi "https://doi.org/10.1016/j.acha.2004.04.001" @default.
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