Heisenberg modules as function spaces
Let Δ Δ be a closed, cocompact subgroup of G * GG× G^, where G is a second countable,
locally compact abelian group. Using localization of Hilbert C^* C∗-modules, we show that
the Heisenberg module E _ Δ (G) E Δ (G) over the twisted group C^* C∗-algebra C^*(Δ, c)
C∗(Δ, c) due to Rieffel can be continuously and densely embedded into the Hilbert space L^
2 (G) L 2 (G). This allows us to characterize a finite set of generators for E _ Δ (G) E Δ (G) as
exactly the generators of multi-window (continuous) Gabor frames over Δ Δ, a result which …
locally compact abelian group. Using localization of Hilbert C^* C∗-modules, we show that
the Heisenberg module E _ Δ (G) E Δ (G) over the twisted group C^* C∗-algebra C^*(Δ, c)
C∗(Δ, c) due to Rieffel can be continuously and densely embedded into the Hilbert space L^
2 (G) L 2 (G). This allows us to characterize a finite set of generators for E _ Δ (G) E Δ (G) as
exactly the generators of multi-window (continuous) Gabor frames over Δ Δ, a result which …
Abstract
Let be a closed, cocompact subgroup of , where G is a second countable, locally compact abelian group. Using localization of Hilbert -modules, we show that the Heisenberg module over the twisted group -algebra due to Rieffel can be continuously and densely embedded into the Hilbert space . This allows us to characterize a finite set of generators for as exactly the generators of multi-window (continuous) Gabor frames over , a result which was previously known only for a dense subspace of . We show that as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if is a lattice, and their associated frame operators corresponding to are bounded.
Springer
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