Sharp Hausdorff-Young inequalities for the quaternion Fourier transforms

Lian, P.

doi:10.1090/proc/14735

Sharp Hausdorff-Young inequalities for the quaternion Fourier transforms
HTML articles powered by AMS MathViewer

by P. Lian PDF

Proc. Amer. Math. Soc. 148 (2020), 697-703 Request permission

Abstract:

The quaternion Fourier transforms are powerful tools in modern data analysis, in particular for color image processing. At present, there are mainly three different quaternion Fourier transforms widely used. In this paper, we prove the sharp Hausdorff-Young inequalities for these three transforms and the more general ones, i.e., the steerable quaternion Fourier transforms. Then Hirschman’s entropy uncertainty principle in the quaternion setting follows from the standard differential approach.

References

K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 531–542 (Russian). MR 0138939
William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
Mawardi Bahri, Eckhard S. M. Hitzer, Ryuichi Ashino, and Rémi Vaillancourt, Windowed Fourier transform of two-dimensional quaternionic signals, Appl. Math. Comput. 216 (2010), no. 8, 2366–2379. MR 2647110, DOI 10.1016/j.amc.2010.03.082
Fred Brackx, Eckhard Hitzer, and Stephen J. Sangwine, History of quaternion and Clifford-Fourier transforms and wavelets, Quaternion and Clifford Fourier transforms and wavelets, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2013, pp. xi–xxvii. MR 3156910
Li-Ping Chen, Kit Ian Kou, and Ming-Sheng Liu, Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform, J. Math. Anal. Appl. 423 (2015), no. 1, 681–700. MR 3273202, DOI 10.1016/j.jmaa.2014.10.003
H. De Bie, Fourier transforms in Clifford analysis, Operator theory edited by Daniel Alpay, Springer Basel, 2015.
Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
Yingxiong Fu, Uwe Kähler, and Paula Cerejeiras, the Balian-low theorem for the windowed quaternionic Fourier transform, Adv. Appl. Clifford Algebr. 22 (2012), no. 4, 1025–1040. MR 2994137, DOI 10.1007/s00006-012-0324-x
Ying Xiong Fu, Non-harmonic quaternion Fourier transform and uncertainty principle, Integral Transforms Spec. Funct. 25 (2014), no. 12, 998–1008. MR 3267753, DOI 10.1080/10652469.2014.961010
Eckhard M. S. Hitzer, Directional uncertainty principle for quaternion Fourier transform, Adv. Appl. Clifford Algebr. 20 (2010), no. 2, 271–284. MR 2645348, DOI 10.1007/s00006-009-0175-2
Eckhard Hitzer, General steerable two-sided Clifford Fourier transform, convolution and Mustard convolution, Adv. Appl. Clifford Algebr. 27 (2017), no. 3, 2215–2234. MR 3688821, DOI 10.1007/s00006-016-0687-5
Eckhard Hitzer and Stephen J. Sangwine, The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations, Quaternion and Clifford Fourier transforms and wavelets, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2013, pp. 15–39. MR 3156912, DOI 10.1007/978-3-0348-0603-9_{2}
Pan Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl. 467 (2018), no. 2, 1258–1269. MR 3842432, DOI 10.1016/j.jmaa.2018.08.002
Todd A. Ell, Nicolas Le Bihan, and Stephen J. Sangwine, Quaternion Fourier transforms for signal and image processing, Focus Series in Digital Signal and Image Processing, John Wiley & Sons, Inc., Hoboken, NJ; ISTE, London, 2014. MR 3308953, DOI 10.1002/9781118930908
R. Wilson and G. H. Granlund, The uncertainty principle in image processing, IEEE Trans. Pattern Anal., 1984, 6(6): 758-767.
Yan Yang and Kit Ian Kou, Novel uncertainty principles associated with 2D quaternion Fourier transforms, Integral Transforms Spec. Funct. 27 (2016), no. 3, 213–226. MR 3455809, DOI 10.1080/10652469.2015.1114482