**ABSTRACT**
### Gabor Systems and the Balian-Low Theorem

#### J.J. Benedetto

###### Department of Mathematics, University of Maryland,College Park, Maryland 20742 andThe MITRE Corporation, McLean, Virginia 22102

C. Heil

###### School of Mathematics, Georgia Institute of Technology,Atlanta, Georgia 30332-0160 andThe MITRE Corporation, Bedford, Massachusetts 01730

D.F. Walnut###### Department of Mathematical Sciences, George Mason University,Fairfax, Virginia 22030

The Balian--Low theorem (BLT) is a key result in time-frequency analysis,
originally stated by Balian and, independently, by Low, as:
If a Gabor system $\{e^{2\pi imbt g(t-na)\}_{m,n \in \Z}$
with $ab=1$ forms an orthonormal basis for $L2$ then
$$(\int_{-\infty}^\infty |t g(t)|^2 dt)
(\int_{-\infty}^\infty |\gamma \hat g(\gamma)|^2 d\gamma
= +\infty$$
The BLT was later extended from orthonormal bases to exact frames.
This paper presents a tutorial on Gabor systems, the BLT, and related topics,
such as the Zak transform and Wilson bases.
Because of the fact that
$(g')^\wedge(\gamma) = 2 \pi i \gamma \hat g(\gamma)$,
the role of differentiation in the proof of the BLT is examined carefully.
We include the construction of a complete
Gabor system of the form $\{see^{2\pi ib_mt} g(t-a_n)\}$ such that
$\{(a_n,b_m)\}$ has density strictly less than 1, and
an Amalgam BLT that provides distinct restrictions on Gabor systems
$\{e^{2\pi imbt} g(t-na)\}$ that form exact frames.