Zak Transforms with Few Zeros and the Tie

A.J.E.M. Janssen

Philips Research Laboratories WY 81, 5656 AA, Eindhoven, The Netherlands
email: A.J.E.M.janssen@philips.com

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.