Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

Abstract

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L 2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in ℝ2 which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ℝ2n with the property that the short-time Fourier transform defines an isometric embedding from L 2(ℝ n ) to L μ 2 (ℝ2n ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.