**ABSTRACT**
### Zak Transforms with Few Zeros and the Tie

#### A.J.E.M. Janssen

##### Philips Research Laboratories WY 81, 5656 AA, Eindhoven, The Netherlands

We consider the difficult problem of deciding whether a triple
$(g,a,b)$, with window $g\in\ldr$ and time shift parameter $a$ and
frequency shift parameter $b$, is a Gabor frame from two different
points of view. We first identify two classes of non-negative
windows $g\in\ldr$ such that their Zak transforms have no and just
one zero per unit square, respectively. The first class consists
of all integrable, non-negative windows $g$ that are supported by and
strictly decreasing on $[0,\infty)$. The second class consists of all even,
non-negative, continuous, integrable windows $g$ that satisfy on
$[0,\infty)$ a condition slightly stronger than strict convexity
(superconvexity). Accordingly, the members of these two classes
generate Gabor frames for integer oversampling factor
$(ab)^{-1}\geq1$ and $\geq\,2$, respectively. When we weaken the
condition of superconvexity into strict convexity, the Zak transforms
$Zg$ may have as
many zeros as one wants, but in all cases $(g,a,b)$ is still a
Gabor frame when $(ab)^{-1}$ is an integer $\geq\,2$. As a second
issue we consider the question for which $a,b>0$ the triple
$(g,a,b)$ is a Gabor frame, where $g$ is the characteristic
function of an interval $[0,c_0)$ with $c_0>0$ fixed. It turns
out that the answer to the latter question is quite complicated,
where irrationality or rationality of $ab$ gives rise to quite
different situations. A pictorial display, in which the various
cases are indicated in the positive $(a,b)$-quadrant, shows a
remarkable resemblance to the design of a low-budget tie.