×

Hyperbolic secants yield Gabor frames. (English) Zbl 1005.42021

It is proved that if \(g(x)= (\pi \gamma/2)^{1/2} (\cosh \pi \gamma t)^{-1}\) for some \(\gamma >0\), then the Gabor system \(\{e^{2\pi imbx}g(x-na)\}_{m,n\in Z}\) is a frame for \(L^2(R)\) for all \(0<ab<1.\) The result is proved by Zak-transform methods, combined with the Ron-Shen criterion for Gabor frames, formulated in the time-domain. In the critical case \(ab=1\) no frame is obtained. Surprisingly, it turns out that the formula for the canonical dual in the limit case \(ab=1\) equals the corresponding limit for the Gaussian.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bastiaans, M. J., Gabor’s expansion of a signal into Gaussian elementary signals, IEEE Proc., 68, 538-539 (1980)
[2] Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 35, 961-1005 (1990) · Zbl 0738.94004
[3] Daubechies, I., Ten Lectures on Wavelets. Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Series in Applied Math. (1992), SIAM: SIAM Philadelphia
[4] (Feichtinger, H. G.; Strohmer, T., Gabor Analysis and Algorithms: Theory and Applications (1998), Birkhäuser: Birkhäuser Boston) · Zbl 0890.42004
[5] Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser: Birkhäuser Boston · Zbl 0966.42020
[6] Janssen, A. J.E. M., Weighted Wigner distributions vanishing on lattices, J. Math. Anal. Appl., 80, 156-167 (1981) · Zbl 0461.46031
[7] Janssen, A. J.E. M., Bargmann transform, Zak transform, and coherent states, J. Math. Phys., 23, 720-731 (1982) · Zbl 0486.46027
[8] Janssen, A. J.E. M., The duality condition for Weyl-Heisenberg frames, (Feichtinger, H. G.; Strohmer, T., Gabor Analysis and Algorithms: Theory and Applications (1998), Birkhäuser: Birkhäuser Boston), 33-84 · Zbl 0890.42006
[9] Ron, A.; Shen, Z., Weyl-Heisenberg frames and Riesz bases in \(L_2(\textbf{R}^d )\), Duke Math. J., 89, 237-282 (1997) · Zbl 0892.42017
[10] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1962), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0105.26901
[11] Zibulski, M.; Zeevi, Y. Y., Oversampling in the Gabor scheme, IEEE Trans. SP, 41, 2679-2687 (1993) · Zbl 0800.94088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.