Zak Transforms with Few Zeros and the Tie

Abstract

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We consider the difficult problem of deciding whether a triple(g,a,b),with window g ∈ L 2(ℝ) and time shift parameter a and frequency shift parameterbis a Gabor frame from two different points of view. We first identify two classes of non-negative windows g ∈ L 2(ℝ) such that their Zak transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, non-negative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, non-negative, continuous, integrable windowsg that satisfy on [0, ∞) a condition slightly stronger than strict convexity (superconvexity). Accordingly, the members of these two classes generate Gabor frames for integer oversampling factor (ab)-1 ≥ 1 and ≥ 2, respectively. When we weaken the condition of superconvexity into strict convexity, the Zak transformsZg 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 ≥ 2. As a second issue we consider the question for which a, b > 0 the triple (g, a, b) is a Gabor frame, where gis the characteristic function of an interval [0, ∞) with c0 > 0 fixed. It turns out that the answer to the latter question is quite complicated, where irrationality or rationality of abgives 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.