# Article

Full entry | PDF   (0.3 MB)
Keywords:
time-frequency concentration; Dunkl-Gabor transform; uncertainty principles
Summary:
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.
References:
[1] Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoam. 19 (2003), 23-55. DOI 10.4171/RMI/337 | MR 1993414
[2] Jeu, M. F. E. de: The Dunkl transform. Invent. Math. 113 (1993), 147-162. DOI 10.1007/BF01244305 | MR 1223227 | Zbl 0789.33007
[3] Demange, B.: Uncertainty principles for the ambiguity function. J. Lond. Math. Soc., II. Ser. 72 (2005), 717-730. DOI 10.1112/S0024610705006903 | MR 2190333 | Zbl 1090.42004
[4] Donoho, D. L., Stark, P. B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49 (1989), 906-931. DOI 10.1137/0149053 | MR 0997928 | Zbl 0689.42001
[5] Dunkl, C. F.: Integral kernels with reflection group invariance. Can. J. Math. 43 (1991), 1213-1227. DOI 10.4153/CJM-1991-069-8 | MR 1145585 | Zbl 0827.33010
[6] Dunkl, C. F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311 (1989), 16-183. DOI 10.1090/S0002-9947-1989-0951883-8 | MR 0951883 | Zbl 0652.33004
[7] Faris, W. G.: Inequalities and uncertainty principles. J. Math. Phys. 19 (1978), 461-466. DOI 10.1063/1.523667 | MR 0484134
[8] Ghobber, S., Jaming, P.: Uncertainty principles for integral orperators. Stud. Math. 220 (2014), 197-220. DOI 10.4064/sm220-3-1 | MR 3173045
[9] Gröchenig, K.: Uncertainty principles for time-frequency representations. Advances in Gabor Analysis H. G. Feichtinger et al. Applied and Numerical Harmonic Analysis Birkhäuser, Basel (2003), 11-30. MR 1955930 | Zbl 1039.42004
[10] Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. 28 Springer, Berlin (1994). MR 1303780
[11] Hogan, J. A., Lakey, J. D.: Time-Frequency and Time-Scale Methods: Adaptive Decompositions, Uncertainty Principles, and Sampling. Applied and Numerical Harmonic Analysis Birkhäuser, Boston (2005). MR 2107799 | Zbl 1079.42027
[12] Lapointe, L., Vinet, L.: Exact operator solution of the Calogero-Sutherland model. Commun. Math. Phys. 178 (1996), 425-452. DOI 10.1007/BF02099456 | MR 1389912 | Zbl 0859.35103
[13] Mejjaoli, H.: Practical inversion formulas for the Dunkl-Gabor transform on $\mathbb R^d$. Integral Transforms Spec. Funct. 23 (2012), 875-890. DOI 10.1080/10652469.2011.647015 | MR 2998902
[14] Mejjaoli, H., Sraieb, N.: Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform. Mediterr. J. Math. 5 (2008), 443-466. DOI 10.1007/s00009-008-0161-2 | MR 2465571 | Zbl 1181.26036
[15] Polychronakos, A. P.: Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69 (1992), 703-705. DOI 10.1103/PhysRevLett.69.703 | MR 1174716 | Zbl 0968.37521
[16] Rösler, M.: An uncertainty principle for the Dunkl transform. Bull. Aust. Math. Soc. 59 (1999), 353-360. DOI 10.1017/S0004972700033025 | MR 1698045 | Zbl 0939.33012
[17] Rösler, M., Voit, M.: Markov processes related with Dunkl operators. Adv. Appl. Math. 21 (1998), 575-643. DOI 10.1006/aama.1998.0609 | MR 1652182 | Zbl 0919.60072
[18] Shimeno, N.: A note on the uncertainty principle for the Dunkl transform. J. Math. Sci., Tokyo 8 (2001), 33-42. MR 1818904 | Zbl 0976.33015
[19] Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math., J. DMV (electronic) 5 (2000), 201-226. MR 1758876 | Zbl 0947.42024

Partner of