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 × B(0, b) in the time-frequency plane ℝd × ℝ̂d. 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.
Similar content being viewed by others
References
A. Bonami, B. Demange, P. Jaming: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoam. 19 (2003), 23–55.
M. F. E. de Jeu: The Dunkl transform. Invent. Math. 113 (1993), 147–162.
B. Demange: Uncertainty principles for the ambiguity function. J. Lond. Math. Soc., II. Ser. 72 (2005), 717–730.
D. L. Donoho, P. B. Stark: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49 (1989), 906–931.
C. F. Dunkl: Integral kernels with reflection group invariance. Can. J. Math. 43 (1991), 1213–1227.
C. F. Dunkl: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311 (1989), 16–183.
W. G. Faris: Inequalities and uncertainty principles. J. Math. Phys. 19 (1978), 461–466.
S. Ghobber, P. Jaming: Uncertainty principles for integral orperators. Stud. Math. 220 (2014), 197–220.
K. Gröchenig: Uncertainty principles for time-frequency representations. Advances in Gabor Analysis (H. G. Feichtinger et al., eds.). Applied and Numerical Harmonic Analysis, Birkhäuser, Basel, 2003, pp. 11–30.
V. Havin, B. Jöricke: The Uncertainty Principle in Harmonic Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 28, Springer, Berlin, 1994.
J. A. Hogan, J. D. Lakey: Time-Frequency and Time-Scale Methods: Adaptive Decompositions, Uncertainty Principles, and Sampling. Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2005.
L. Lapointe, L. Vinet: Exact operator solution of the Calogero-Sutherland model. Commun. Math. Phys. 178 (1996), 425–452.
H. Mejjaoli: Practical inversion formulas for the Dunkl-Gabor transform on ℝd. Integral Transforms Spec. Funct. 23 (2012), 875–890.
H. Mejjaoli, N. Sraieb: Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform. Mediterr. J. Math. 5 (2008), 443–466.
A. P. Polychronakos: Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69 (1992), 703–705.
M. Rösler: An uncertainty principle for the Dunkl transform. Bull. Aust. Math. Soc. 59 (1999), 353–360.
M. Rösler, M. Voit: Markov processes related with Dunkl operators. Adv. Appl. Math. 21 (1998), 575–643.
N. Shimeno: A note on the uncertainty principle for the Dunkl transform. J. Math. Sci., Tokyo 8 (2001), 33–42.
E. Wilczok: New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math., J. DMV (electronic) 5 (2000), 201–226.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghobber, S. Dunkl-Gabor transform and time-frequency concentration. Czech Math J 65, 255–270 (2015). https://doi.org/10.1007/s10587-015-0172-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-015-0172-7