On generating tight Gabor frames at critical density

We consider the construction of tight Gabor frames (h, a = 1, b = 1) from Gabor systems (g, a = 1, b = 1) with g a window having few zeros in the Zak transform domain via the operation h = Z^-1 (Zg/|Zg|), where Z is the standard Zak transform. We consider this operation with g the Gaussian, the hyperbolic secant, and for g belonging to a class of positive, even, unimodal, rapidly decaying windows of which the two-sided exponential is a typical example. All these windows g have the property that Zg has a single zero, viz. at (1/2, 1/2), in the unit square [0,1)². The Gaussian and hyperbolic secant yield a frame for any a, b > 0 with ab < 1, and we show that so does the two-sided exponential. For these three windows it holds that S_a^-1/2 g ¿ h as a ¿ 1, where S_a is the frame operator corresponding to the Gabor frame (g, a, a). It turns out that the h's corresponding to g's of the above type look and behave quite similarly when scaling parameters are set appropriately. We give a particular detailed analysis of the h corresponding to the two-sided exponential. We give several representations of this h, and we show that h ¿ L¹ (R) n L ⁸ (R), and is continuous and differentiable everywhere except at the half-integers, etc., and we pay particular attention to the cases that the time constant of the two-sided exponential g tends to 8. We also consider the cases that the time constants of the Gaussian and of the hyperbolic secant tend to 0 or to 8. It so turns out that h thus obtained changes from the box function ¿_(-1/2,1/2) into Fourier transform sine p · when the time constant changes from 0 to 8.