Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. (English) Zbl 1425.42012
Summary: The quaternion Fourier transform – a generalized form of the classical Fourier transform - has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt’s inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form \(f=f_1+\mathbf{i}f_2+f_3\mathbf{j}+\mathbf{i}f_4\mathbf{j}\) of quaternion from Hitzer and the novel module or \(L^p\)-norm of the quaternion Fourier transform \(\hat{f}\), then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg-Weyl’s uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt’s inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
Keywords:
quaternion Fourier transform; Pitt’s inequality; logarithmic uncertainty estimate; uncertainty principleReferences:
| [1] | Aytur, O.; Ozaktas, H. M., Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform, Opt. Commun., 120, 166-170 (1995) |
| [2] | Bahri, M.; Hitzer, E.; Hayashi, A.; Ashino, R., An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl., 56, 2398-2410 (2008) · Zbl 1165.42310 |
| [3] | Bas, P.; Le Bihan, N.; Chassery, J. M., Color image watermarking using quaternion Fourier transform, (Proceedings of the IEEE International Conference on Acoustics Speech and Signal and Signal Processing. Proceedings of the IEEE International Conference on Acoustics Speech and Signal and Signal Processing, ICASSP, Hong Kong (2003)), 521-524 |
| [4] | Bayro-Corrochano, E.; Trujillo, N.; Naranjo, M., Quaternion Fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations, J. Math. Imaging Vision, 28, 2, 179-190 (2007) |
| [5] | Beckner, W., Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123, 1897-1905 (1995) · Zbl 0842.58077 |
| [6] | Beckner, W., Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136, 1871-1885 (2008) · Zbl 1221.42016 |
| [7] | Cohen, L., The uncertainty principles of windowed wave functions, Opt. Commun., 179, 221-229 (2000) |
| [8] | Da, Z. X., Modern Signal Processing, 362 (2002), Tsinghua University Press: Tsinghua University Press Beijing |
| [9] | Dang, P.; Deng, G. T.; Qian, T., A sharper uncertainty principle, J. Funct. Anal., 265, 2239-2266 (2013) · Zbl 1284.42019 |
| [10] | Dembo, A.; Cover, T. M., Information theoretic inequalities, IEEE Trans. Inform. Theory, 37, 6, 1501-1508 (1991) · Zbl 0741.94001 |
| [11] | Donoho, D. L.; Stark, P. B., Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49, 3, 906-931 (1989) · Zbl 0689.42001 |
| [12] | Fefferman, Charles L., The uncertainty principle, Bull. Amer. Math. Soc. (N.S.), 9, 2 (September 1983) · Zbl 0526.35080 |
| [13] | Folland, G. B.; Sitaram, A., The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3, 207-238 (1997) · Zbl 0885.42006 |
| [14] | Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0966.42020 |
| [15] | Hardy, G.; Littlewood, J. E.; Polya, G., Inequalities (1951), Cambridge University Press · Zbl 0634.26008 |
| [16] | Heinig, H.; Smith, M., Extensions of the Heisenberg-Weyl inequality, Int. J. Math. Math. Sci., 9, 185-192 (1986) · Zbl 0589.42009 |
| [17] | Hitzer, E., Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr., 17, 3, 497-517 (2007) · Zbl 1143.42006 |
| [18] | Hitzer, E., Directional uncertainty principle for quaternion Fourier transform, Adv. Appl. Clifford Algebr., 20, 271-284 (2010) · Zbl 1198.42006 |
| [19] | Iwo, B. B., Entropic uncertainty relations in quantum mechanics, (Accardi, L.; Von Waldenfels, W., Quantum Probability and Applications II. Quantum Probability and Applications II, Lecture Notes in Math., vol. 1136 (1985), Springer: Springer Berlin), 90 · Zbl 0584.94009 |
| [20] | Iwo, B. B., Formulation of the uncertainty relations in terms of the Rényi entropies, Phys. Rev. A, 74, 052101 (2006) |
| [21] | Iwo, B. B., Rényi entropy and the uncertainty relations, (Adenier, G.; Fuchs, C. A.; Khrennikov, A. Yu, Foundations of Probability and Physics. Foundations of Probability and Physics, AIP Conf. Proc., vol. 889 (2007), American Institute of Physics: American Institute of Physics Melville), 52-62 · Zbl 1139.81302 |
| [22] | Kou, K. I.; Ou, J.; Morais, J., On uncertainty principle for quaternionic linear canonical transform, Abstr. Appl. Anal., 2013 (2013), Article ID 725952, 14 pp · Zbl 1275.42012 |
| [23] | Lieb, E.; Loss, M., Analysis (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0966.26002 |
| [24] | Loughlin, P. J.; Cohen, L., The uncertainty principle: global, local, or both?, IEEE Trans. Signal Process., 52, 5, 1218-1227 (2004) · Zbl 1370.94181 |
| [25] | Maassen, H., A discrete entropic uncertainty relation, (Quantum Probability and Applications. Quantum Probability and Applications, Lecture Notes in Math. (1990), Springer: Springer Berlin, Heidelberg), 263-266 · Zbl 0725.46039 |
| [26] | Maassen, H.; Uffink, J. B.M., Generalized entropic uncertainty relations, Phys. Rev. Lett., 60, 12, 1103-1106 (1988) |
| [27] | Majernik, V.; Eva, M.; Shpyrko, S., Uncertainty relations expressed by Shannon-like entropies, CEJP, 3, 393-420 (2003) |
| [28] | Mawardi, B.; Hitzer, E.; Hayashi, A.; Ashino, R., An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl., 56, 9, 2411-2417 (2008) |
| [29] | Mustard, D., Uncertainty principle invariant under fractional Fourier transform, J. Austral. Math. Soc. Ser. B, 33, 180-191 (1991) · Zbl 0765.43004 |
| [30] | Nicewarner, K. E.; Sanderson, A. C., A general representation for orientational uncertainty using random unit quaternions, (Proc. IEEE International Conference on Robotics and Automation (1994)), 1161-1168 |
| [31] | Ozaktas, H. M.; Aytur, O., Fractional Fourier domains, Signal Process., 46, 119-124 (1995) · Zbl 0875.94037 |
| [32] | Rényi, A., On measures of information and entropy, (Proc. Fourth Berkeley Symp. on Mathematics, Statistics and Probability (1960)), 547 |
| [33] | Sangwine, S. J.; Ell, T. A., Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process., 16, 1, 22-35 (2007) · Zbl 1279.94014 |
| [34] | Shinde, S.; Vikram, M. G., An uncertainty principle for real signals in the fractional Fourier transform domain, IEEE Trans. Signal Process., 49, 11, 2545-2548 (2001) · Zbl 1369.94287 |
| [35] | Stern, A., Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Amer. A, 25, 3, 647-652 (2008) |
| [36] | Wódkiewicz, K., Operational approach to phase-space measurements in quantum mechanics, Phys. Rev. Lett., 52, 13, 1064-1067 (1984) |
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.
