Hyperbolic secants yield Gabor frames

By Augustus J. E. M. Janssen, Thomas Strohmer. 2002

98 citations recorded by the source corpus.

Abstract We show that ( g 2 , a , b ) is a Gabor frame when a >0, b >0, ab g 2 (t)=( 1 2 πγ) 1/2 (cosh πγt) −1 is a hyperbolic secant with scaling parameter γ>0. This is accomplished by expressing the Zak transform of g 2 in terms of the Zak transform of the Gaussian g 1 ( t )=(2γ) 1/4 exp (−πγ t 2 ), 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 g 2 and g 1 are the same at critical density a = b =1. Also, we display the “singular” dual function corresponding to the hyperbolic secant at critical density.

Subfields: Applied Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/1ab72b9991b39b860df7ecd192fbe9eede5377e0

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Citations in the corpus

Gabor Frames and Totally Positive Functions (2011, methodology): “A general method for constructing functions in M can be extracted from [22].”
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems (2005, background): “Such a complete characterization of frame properties is known for only a few speciﬁc functions [Jan94], [Jan03], [JS02], cf. also [CK02].”
History and Evolution of the Density Theorem for Gabor Frames (2007, unknown): “(b) Janssen and Strohmer [131] proved that for the hyperbolic secant window g(t) = (π/2) 1/2 (cosh πt) −1 , G(g, αZ × βZ) is a frame if and only if 0 αβ 1.”
Gabor (super)frames with Hermite functions (2008, background): “, a > 0, in [L, SW], for the hyperbolic secant g(t) = (cosh at) −1 i n [JS], and for the one-sided exponential function g(t) = e −a|t| χ[0,∞)(t) [J3].”
Finite Discrete Gabor Analysis (2007, methodology): “For more details on the hyperbolic secant as a Gabor window, see [59].”
The Mystery of Gabor Frames (2014, methodology): “The case of the hyperbolic secant (ii) was reduced to the case of the Gaussian by Janssen and Strohmer [60], and the two exponential functions in (ii) and (iii) were analyzed by Janssen [56,58].”
Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions (2009, background): “See the survey article [28], [13] and the important special windows studied in [35].”