On the Weyl-Heisenberg frames generated by simple functions

Let with , and let (?,a,1), 0<a?1 be a Weyl-Heisenberg system {e2πimx?(x−na):m,n∈Z}. We show that if E=0,1] (and some modulo extension of E), then (?,a,1) is a frame for each 0<a?1 (for certain a, respectively) if and only if the analytic function has no zero on the unit circle {z:|z|=1}. These results extend the case of Casazza and Kalton (2002) 6] that and a=1, which brought together the frame theory and the function theory on the closed unit disk. Our techniques of proofs are based on the Zak transform and the distribution of fractional parts of {na}_n∈Z.