Hyperbolic secants yield Gabor frames
نویسنده
چکیده
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 follows that the windows, generating tight Gabor frames, that are canonically associated to g2 and g1 are the same at critical density a = b = 1. Also, we display the “singular” dual function corresponding to the hyperbolic secant at critical density. AMS Subject Classification: 42C15, 33D10, 94A12.
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