The Zak transform and some counterexamples in time-frequency analysis

It is shown how the Zak transform can be used to find nontrivial examples of functions f, g E L 2 ( 2 ) with f . g = 0 = F . C, where F, G are the Fourier transforms of f, g, respectively. This is then used to exhibit a nontrivial pair of functions h, keL2(T1), h # k, such that I h I = 1 k 1 , I H I = I K 1 . A similar construction is used to find an abundance of nontrivial pairs of functions h, k6L2(F2), h # k , with I A , I = I A , I or with I W, I = 1 W , 1 , where A,, A , and W,, W, are Manuscript received July 12, 1988; revised July 2, 1991. The author is with Philips Research Laboratories, WAY-2, 5600 JA IEEE Log Number 9103458. Eindhoven, The Netherlands. 0018-9448/92$03.00