Compactly supported multiwindow dual Gabor frames of rational sampling density

Abstract

We consider multiwindow Gabor systems (G (N) ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski-Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski-Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a = 1 and b = 1. As another application, we show that a multiwindow Gabor system (G (N) ; 1, 1) forms an orthonormal basis if and only if it has only one window (N = 1) which is a sum of characteristic functions whose supports 'essentially' form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4).