Zak transforms with few zeros and the tie. (English) Zbl 1027.42025

Janssen, A. J. E. M.

Feichtinger, Hans G. (ed.) et al., Advances in Gabor analysis. Basel: Birkhäuser. Applied and Numerical Harmonic Analysis. 31-70 (2003).

Summary: We consider the difficult problem of deciding whether a triple \((g, a, b)\), with window \(g\in L^2(\mathbb{R})\) 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 nonnegative windows \(g\in L^2(\mathbb{R})\) such that their Zak transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, nonnegative windows \(g\) that are supported by and strictly decreasing on \([0,\infty)\). The second class consists of all even, nonnegative, 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 \((ab)^{-1}\geq 1\) 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.
For the entire collection see [Zbl 1005.00015].

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15	General harmonic expansions, frames

Keywords:

nonnegative windows; Zak transforms; Gabor frames; oversampling factor; superconvexity; convexity; irrationality; rationality; low-budget tie