Abstract
For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg’s inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.
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References
Bahri, M.: A modified uncertainty principle for two sided quaternion Fourier transform. Adv. Appl. Clifford Algebr. 26, 513–527 (2016)
Bahri, M., Ashino, R., Vaillancourt, R.: Two-dimensional quaternion wavelet transform. Applied Mathematics and Computation 218, 10–21 (2011)
Bahri, M., Hitzer, E., Hayashi, A., Ashino, R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl 56, 2398–2410 (2008)
Benedicks, M.: On Fourier transforms of function supported on sets of finite Lebesgue measure. J. Math. Anal. Appl 106, 180–183 (1985)
Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math 49(3), 906–931 (1989)
Gupta,B., Verma,A.K., Cattani,C.:A new class of linear canonical wavelet transform. Journal of Applied and Computational Mechanics, (2023) https://doi.org/10.22055/jacm.2023.44326.4196 [In press]
Hardy, G.H.: A theorem concerning Fourier transform. J. London Math. Soc 8, 227–231 (1933)
Havin,V., and Jöricke,B.: The Uncertainty Principle in Harmonic Analysis. Ergebnisse der Mathematik undihrer Grenzgebiete 3. Folge. 28, Springer, Berlin, (1994)
Heisenberg,W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik, Zeit. Phys;43,172, The Physical Principles of the Quantum Theory (Dover, New York, 1949) (The University of Chicago Press, 1930) (1927)
Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv Appl Cliff Alg 17(3), 497–517 (2007)
Hitzer, E.: Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebr 20, 271–284 (2010)
Hleili, K.: Uncertainty principles for spherical mean \(L^2\)-multiplier operators. J Pseudo-Differ Oper Appl 9, 573–587 (2018)
Hleili, K.: Continuous wavelet transform and uncertainty principle related to the Weinstein operator. Integral Transf Spec Funct 29(4), 252–268 (2018)
Hleili, K.: Some Results for the Windowed Fourier Transform Related to the spherical mean Operator. Acta Mathematica Vietnamica. 46(1), 179–201 (2021)
Hleili, K.: A variety of uncertainty principles for the Hankel-Stockwell transform. Open journal of Mathematical analysis 5(1), 22–34 (2021)
Hleili, K.: A Variation on Uncertainty Principles for Quaternion Linear Canonical Transform. Advances in Applied Clifford Algebras 31(3), 1–13 (2021)
Hleili, K.: \(L^p\) uncertainty principles for the Windowed spherical mean transform. Memoirs on Differential Equations and Mathematical Physics 85, 75–90 (2022)
Hleili, K.: Reproducing inversion formulas and uncertainty principles for the windowed Fourier transform related to the Spherical mean operator. Asian-European Journal of Mathematics 16(09), 2350159 (2023)
Kaur, N., Gupta, B., Verma, A.K.: Multidimensional fractional wavelet transforms and uncertainty principles. Journal of Computational and Applied Mathematics. 430, 115250 (2023)
Kou,K.I., Ou,J., Morais,J.: On uncertainty principle for quaternionic linear canonical transform, Abstr. Appl. Anal,(2013), https://doi.org/10.1155/2013/725952, Article ID 725952, 14 pp (2013)
Lian,P.: Uncertainty principle for the quaternion Fourier transform. J Math Anal Appl, 467(2), 1258–1269. MR 3842432, https://doi.org/10.1016/j.jmaa.2018.08.002. (2018)
Li-Ping,C., Kit Ian Kou,K., Ming-Sheng,L.: Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J. Math. Anal. Appl, 423, 681–700(2015)
Mawardi, B., Hitzer, E., Hayashi, A., Ashino, R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl 56(9), 2411–2417 (2008)
Morgan,G.W.: A note on Fourier transforms. J. London Math. Soc, 9,188–192(1934)
Price,J.F.: Inequalities and local uncertainty principles. J. Math. Phys, 24, 1711–1714(1983)
Saitoh, S.: Theory of Reproducing Kernels and Its Application. Pitman Research Notes in Mathematics Series. Longman Higher Education, London (1988)
Su, Y.: Heisenberg type uncertainty principle for continuous shearlet transform. J Nonlinear Sci Appl 9, 778–786 (2016)
Tefjeni,E., Brahim,K.: Uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform. Integr. Transf. Spec. F,31(8), 669–684(2020)
Wilczok, E.: New uncertainty principles for the continuous gabor transform and the continuous wavelet transform. Doc. Math. 5, 201–226 (2000)
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Hleili, M. Some Uncertainty Principles for the Right-Sided Multivariate Continuous Quaternion Wavelet Transform. Adv. Appl. Clifford Algebras 34, 13 (2024). https://doi.org/10.1007/s00006-024-01319-w
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DOI: https://doi.org/10.1007/s00006-024-01319-w
Keywords
- Multivariate quaternion Fourier transform
- The multivariate continuous quaternion wavelet transform
- Signal recovery
- Heisenberg–Pauli–Weyl inequality
- Local uncertainty principle