Hyperbolic secants yield Gabor frames
| dc.creator | Janssen, A. J. E. M. | |
| dc.creator | Strohmer, Thomas | |
| dc.date | 2003-01-13 | |
| dc.date.accessioned | 2026-07-07T04:54:26Z | |
| dc.date.available | 2026-07-07T04:54:26Z | |
| dc.description | We show that $(g_2,a,b)$ is a Gabor frame when $a>0, b>0, ab<1$ and $g_2(t)=({1/2}πγ)^{1/2} (\cosh πγt)^{-1}$ 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 $g_1(t)=(2γ)^{1/4} \exp (-πγt^2)$, 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. | |
| dc.identifier | https://arxiv.org/abs/math/0301134 | |
| dc.identifier | http://arxiv.org/abs/math/0301134 | |
| dc.identifier | Appl. Comp. Harm. Anal., 12(2): 259--267, 2002 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66249 | |
| dc.subject | Functional Analysis | |
| dc.title | Hyperbolic secants yield Gabor frames | |
| dc.type | text |