Abstract:
Let G be a locally compact abelian group, and let ∆ be a discrete, cocompact subgroup of G × G. The Gabor system generated by η ∈ L2(G) over ∆ is the set of functions {ηx,ω : (x, ω) ∈ ∆} where
\(\quad\eta_{x,\omega}(t) = \omega(t) \eta(t-x)\)
for t ∈ G. The following problem remains unsolved: Given ∆, does there exist a function η ∈ L2(G) such that {ηx,ω : (x,ω) ∈ ∆} is a Gabor frame? Here, a frame is a type of well-behaved generating set for L2(G) that generalizes orthonormal bases.
The definition of a Gabor system is reminiscent of the Heisenberg representation associated to the group G. Indeed, aspects of the theory of Gabor frames have been shown to have a close relationship to a construction of M. Rieffel known as Heisenberg modules. Given ∆, the Heisenberg module ℰ∆(G) is a left Hilbert C*-module over the twisted group C*-algebra C*(∆,c).
M. Jakobsen and F. Luef showed that there exists a Gabor frame over ∆ with a “nice” generator η if and only if the Heisenberg module is singly generated. In this way, a problem about Gabor frames can be transferred into the domain of operator algebras. Indeed, this method has been used to prove new results about the existence and nonexistence of Gabor frames. In this talk, I will present some of my work on an application of Heisenberg modules to the Balian–Low theorem. This is a theorem about the nonexistence of Gabor frames in certain situations and is related to the uncertainty principle.