Dunkl-Gabor transform and time-frequency concentration

Saifallah Ghobber

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 255-270
  • ISSN: 0011-4642

Abstract

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The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg’s uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks’ uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with respect to a particular window function cannot be time-frequency concentrated in a subset of the form S × ℬ ( 0 , b ) in the time-frequency plane ℝ d × ℝ ^ d . As a side result we generalize a related result of Donoho and Stark on stable recovery of a signal which has been truncated and corrupted by noise.

How to cite

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Ghobber, Saifallah. "Dunkl-Gabor transform and time-frequency concentration." Czechoslovak Mathematical Journal 65.1 (2015): 255-270. <http://eudml.org/doc/270039>.

@article{Ghobber2015,
abstract = {The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg’s uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks’ uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with respect to a particular window function cannot be time-frequency concentrated in a subset of the form $S\times \mathcal \{B\}(0,b)$ in the time-frequency plane $\mathbb \{R\}^d\times \widehat\{\mathbb \{R\}\}^d$. As a side result we generalize a related result of Donoho and Stark on stable recovery of a signal which has been truncated and corrupted by noise.},
author = {Ghobber, Saifallah},
journal = {Czechoslovak Mathematical Journal},
keywords = {time-frequency concentration; Dunkl-Gabor transform; uncertainty principles},
language = {eng},
number = {1},
pages = {255-270},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Dunkl-Gabor transform and time-frequency concentration},
url = {http://eudml.org/doc/270039},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Ghobber, Saifallah
TI - Dunkl-Gabor transform and time-frequency concentration
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 255
EP - 270
AB - The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg’s uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks’ uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with respect to a particular window function cannot be time-frequency concentrated in a subset of the form $S\times \mathcal {B}(0,b)$ in the time-frequency plane $\mathbb {R}^d\times \widehat{\mathbb {R}}^d$. As a side result we generalize a related result of Donoho and Stark on stable recovery of a signal which has been truncated and corrupted by noise.
LA - eng
KW - time-frequency concentration; Dunkl-Gabor transform; uncertainty principles
UR - http://eudml.org/doc/270039
ER -

References

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