Hyperbolic Secants Yield Gabor Frames

Hyperbolic secants yield Gabor frames

We show that (g2, a, b) is a Gabor frame when a > 0, b > 0, ab < 1 and g2(t) = ( 1 2πγ) 1 2 (cosh πγt)−1 is a hyperbolic secant with scaling parameter γ > 0. This is accomplished by expressing the Zak transform of g2 in terms of the Zak transform of the Gaussian g1(t) = (2γ) 1 4 exp(−πγt2), together with an appropriate use of the Ron-Shen criterion for being a Gabor frame. As a side result it f...

Nonstationary Gabor Frames

To overcome the limitation induced by the fixed timefrequency resolution over the whole time-frequency plane of Gabor frames, we propose a simple extension of the Gabor theory leading to the construction of frames with timefrequency resolution evolving over time or frequency. We describe the construction of such frames and give the explicit formulation of the canonical dual frame for some condi...

Density of Gabor Frames

Image processing by alternate dual Gabor frames

‎We present an application of the dual Gabor frames to image‎ ‎processing‎. ‎Our algorithm is based on finding some dual Gabor‎ ‎frame generators which reconstructs accurately the elements of the‎ ‎underlying Hilbert space‎. ‎The advantages of these duals‎ ‎constructed by a polynomial of Gabor frame generators are compared‎ ‎with their canonical dual‎.

Gabor Frames without Inequalities

We prove fourteen equivalent conditions for a set of time-frequency shifts on a lattice Λ, {e2g(t − λ1) : (λ1, λ2) ∈ Λ} ⊆ L (R), to be a frame for L(R). Remarkably, several of these conditions can be formulated without an inequality. In particular, instead of checking the invertibility of the frame operator on L(R), it suffices to verify that it is one-to-one on a certain subspace of tempered d...