On the Weyl–Heisenberg frames generated by simple functions

Let φ(x) = ∞ n=0 cnχE(x − n) with {cn}∞ n=0 ∈ l1, and let (φ, a, 1), 0 < a 1 be a Weyl–Heisenberg system {e2πimxφ(x − na): m,n ∈ Z}. We show that if E = [0, 1] (and some modulo extension of E), then (φ, a, 1) is a frame for each 0 < a 1 (for certain a, respectively) if and only if the analytic function H(z) = ∞ n=0 cnzn has no zero on the unit circle {z: |z| = 1}. These results extend the case of Casazza and Kalton (2002) [6] that φ(x) = k i=1 χ[0,1](x −ni ) 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. © 2011 Elsevier Inc. All rights reserved.