On Benedicks–Amrein–Berthier uncertainty principles for continuous quaternion wavelet transform
Xinyu Wang
Department of Mathematics, Beijing Jiaotong University, Beijing, China
Contribution: Formal analysis, Conceptualization, Writing - original draft, Investigation, Methodology
Search for more papers by this authorCorresponding Author
Shenzhou Zheng
Department of Mathematics, Beijing Jiaotong University, Beijing, China
Correspondence
Shenzhou Zheng, Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China.
Email: [email protected]
Communicated by: Z. Li
Contribution: Methodology, Conceptualization, Investigation, Formal analysis, Supervision, Funding acquisition, Writing - review & editing, Validation, Project administration, Writing - original draft
Search for more papers by this authorXinyu Wang
Department of Mathematics, Beijing Jiaotong University, Beijing, China
Contribution: Formal analysis, Conceptualization, Writing - original draft, Investigation, Methodology
Search for more papers by this authorCorresponding Author
Shenzhou Zheng
Department of Mathematics, Beijing Jiaotong University, Beijing, China
Correspondence
Shenzhou Zheng, Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China.
Email: [email protected]
Communicated by: Z. Li
Contribution: Methodology, Conceptualization, Investigation, Formal analysis, Supervision, Funding acquisition, Writing - review & editing, Validation, Project administration, Writing - original draft
Search for more papers by this authorAbstract
The continuous quaternion wavelet transform (CQWT) can refine a quaternion function in the multiscale framework by stretching and translation to achieve an effect of the localized analysis. In this paper, we are devoted to some different types of uncertainty principles (UPs) for the two-dimensional CQWT. More precisely, we obtain the Benedicks–Amrein–Berthier UP and the logarithmic Sobolev-type UP for the CQWT. As a direct consequence, we also deduce some significant corollaries, such as the Benedicks UP, the general Heisenberg-type UP, the general concentration UP, and the concentration logarithmic Sobolev-type UP for the CQWT.
CONFLICT OF INTEREST STATEMENT
The authors declare that they have no conflict of interest in this article.
REFERENCES
- 1A. Grossman and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723–736.
- 2R. Murenzi, Wavelet transforms associated to the
-dimensional Euclidean group with dilations, Wavelet, time-frequency methods and phase space. Jones and Bartlett, Publishers, Boston, 1989, pp. 239–246.
10.1007/978-3-642-97177-8_22 Google Scholar
- 3L. Traversoni, Imaging analysis using quaternion wavelet, geometric algebra with applications in science and engineering, Birkhäuser, Boston, 2001, pp. 326–345.
- 4J. Zhao and L. Peng, Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations, J. Nat. Geom. 20 (2001), no. 1/2, 21–32.
- 5P. Chai, X. Luo, and Z. Zhang, Image fusion using quaternion wavelet transform and multiple features, IEEE Access 5 (2017), 6724–6734.
- 6L. Fan, Y. Wang, and H. Zhang, High-accuracy 3D contour measurement by using the quaternion wavelet transform image denoising technique, Electronics 11 (2022), no. 12, 1807.
- 7P. Niu, L. Wang, and X. Shen, A novel robust image watermarking in quaternion wavelet domain based on superpixel segmentation,Multidim. Syst. Signal Pr. 31 (2020), no. 4, 1509–1530.
- 8J. Wang, T. Li, and X. Luo, Identifying computer generated images based on quaternion central moments in color quaternion wavelet domain, IEEE, Trans. Circ. Syst. Vid. 29 (2018), no. 9, 2775–2785.
- 9M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180–183.
- 10W. O. Amrein and A. M. Berthier, On support properties of -functions and their Fourier transforms, J. Funct. Fnal. 24 (1977), no. 3, 258–267.
- 11F. L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, St. Petersburg Math. J. 5 (1994), no. 4, 663–717.
- 12P. Jaming, Nazarov's uncertainty principles in higher dimension, J. Approx Theory 149 (2007), no. 1, 30–41.
- 13S. Ghobber and P. Jaming, Strong annihilating pairs for the Fourier-Bessel transform, J. Math. Anal. Appl. 377 (2011), no. 2, 501–515.
- 14S. Ghobber and S. Omri, Time-frequency concentration of the windowed Hankel transform, Integral Transform Spec. Funct. 25 (2014), no. 6, 481–496.
- 15A. Achak and R. Daher, Benedicks-Amrein-Berthier type theorem related to Weinstein transform, Anal. Math. 43 (2017), no. 4, 511–521.
- 16A. Achak and R. Daher, Benedicks-Amrein-Berthier type theorem related to Opdam-Cherednik transform, J. Pseudo Differ. Oper. Appl. 9 (2018), no. 2, 431–441.
- 17X. Zhu and S. Zheng, Uncertainty principles for the two-sided offset quaternion linear canonical transform, Math. Meth. Appl. Sci. 44 (2021), 14236–14255.
- 18M. Zayed and Y. El Haoui, The uncertainty principle for the octonion Fourier transform, Math. Meth. Appl. Sci. 46 (2023), 2651–2666.
- 19F. Alhajji and S. Ghobber, Some uncertainty inequalities for the continuous wavelet transform, J. Pseudo. Differ. Oper. Appl. 12 (2021), no. 2, 1–25.
- 20O. Tyr and R. Daher, Benedicks-Amrein-Berthier type theorem and local uncertainty principles in Clifford algebras, Rendiconti del Circolo Matematico di Palermo Series 72 (2023), 99–115.
- 21Y. El Haoui and M. Zayed, A new uncertainty principle related to the generalized quaternion Fourier transform, J. Pseudo Differ. Oper. Appl. 12 (2021), no. 4, 1–13.
- 22H. Kubo, T. Ogawa, and T. Suguro, Beckner type of the logarithmic Sobolev and a new type of Shannon's inequalities and an application to the uncertainty principle, Proc. Am. Math. Soc. 147 (2019), no. 4, 1511–1518.
- 23W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Am. Math. Soc. 123 (1995), no. 6, 1897–1905.
- 24H. Mejjaoli and N. Sraieb, New uncertainty principles for the Dunkl wavelet transform, Int. J. Open Problems Complex Anal. 12 (2020), no. 2, 51–75.
- 25M. Bahri, F. A. Shah, and A. Y. Tantary, Uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions, Integral Transform Spec. Funct. 31 (2020), no. 7, 538–555.
- 26H. Mejjaoli and F. A. Shah, Uncertainty principles associated with the directional short-time Fourier transform, J. Math. Phys. 62 (2021), no. 6, 063511.
- 27T. Suguro, Shannon's inequality for the Rényi entropy and an application to the uncertainty principle, J. Funct. Anal. 283 (2022), no. 6, 109566.
- 28S. Ghobber, Sharp Shannon's and logarithmic Sobolev inequalities for the Hankel transform, J. Math. Anal. Appl. 507 (2022), no. 1, 125740.
- 29J. Voight, Quaternion algebras, graduate texts in mathematics, Springer Cham, 2021.
10.1007/978-3-030-56694-4 Google Scholar
- 30E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr. 17 (2007), no. 3, 497–517.
- 31E. Hitzer, Directional uncertainty principle for quaternion Fourier transform, Adv. Appl. Clifford Algebr. 20 (2010), no. 2, 271–284.
- 32P. Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl. 467 (2018), no. 2, 1258–1269.
- 33M. Bahri and R. Ashino, A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms, Abstr. Appl. Anal. 2017 (2017), 1–11.
10.1155/2017/3795120 Google Scholar
- 34T. A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, Proceedings of 32nd I,EEE Conference on Decision and Control, 1993, pp. 1830–1841.
- 35S. C. Pei, J. J. Ding, and J. H. Chang, Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT, IEEE, Trans. Signal Process. 49 (2001), no. 11, 2783–2797.
- 36T. Bülow. (1999). Hypercomplex spectral signal representations for the processing and analysis of images, Thesis PhD, Christian-Albrechts-Universität.
- 37Y. El Haoui, The continuous quaternion algebra-valued wavelet transform and the associated uncertainty principle, J. Pseudo Differ. Oper. Appl. 12 (2021), no. 1, 1–23.
- 38M. Bahri, R. Ashino, and R. Vaillancourt, Two-dimensional quaternion wavelet transform, Appl. Math. Comput. 218 (2011), no. 1, 10–21.
- 39C. Baccar, Uncertainty principles for the continuous Hankel wavelet transform, Integral Transform Spec. Funct. 27 (2016), no. 6, 413–429.
- 40X. Wang and S. Zheng, Tighter Heisenberg-Weyl type uncertainty principle associated with quaternion wavelet transform, J. Pseudo-Differ. Oper. Appl. 14 (2023), no. 1, 1–27.
- 41K. Trimeche, Generalized harmonic analysis and wavelet packets: an elementary treatment of theory and applications, CRC Press, London, 2001.
