Frame set for Gabor systems with Haar window

Abstract

We describe the full structure of the frame set for the Gabor system G(g;α,β):={e−2πimβ⋅g(⋅−nα):m,n∈Z} with the window being the Haar function g=−χ[−1/2,0)+χ[0,1/2). This is the first compactly supported window function for which the frame set is represented explicitly.The strategy of this paper is to introduce the piecewise linear transformation M on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system G(g;α,β) to be a frame, i.e., the symmetric invariant set of the transformation M is empty.