A decay result for certain windows generating orthogonal Gabor bases

We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z ^-1(Zg/|Zg|). Here Z is the standard Zak transform and g is an even, real, well-behaved window such that Zg has exactly one zero, at 1/2,1/2, in [0,1)². We show that h and its Fourier transform have maximal decay as allowed by the Balian-Low theorem. Our result illustrates a theorem of Benedetto, Czaja, Gadzinski, and Powell, case p=q=2, on sharpness of the Balian-Low theorem.