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Heisenberg’s and Hardy’s uncertainty principles for special relativistic space-time Fourier transformation. (English) Zbl 1454.15017

In [Adv. Appl. Clifford Algebr. 17, No. 3, 497–517 (2007; Zbl 1143.42006)], E. M. S. Hitzer generalized the notion of quaternion Fourier transform (QFT) to a new noncommutative multivector Fourier transform, the so-called space-time Fourier transform (SFT), of functions from the spacetime \(\mathbb{R}^{3,1}\) to \(Cl_{(3,1)}\).
In this paper, the authors investigate SFT further and give several important properties such as Plancherel’s theorem, the Hausdorff-Young inequality, uniform continuity, and the Riemann-Lebesgue lemma. Using the space-time split (\(\pm\) split of the space-time algebra), they prove the Heisenberg uncertainty principle and derive Hardy’s theorem related to the SFT.

MSC:

15A67 Applications of Clifford algebras to physics, etc.
83A05 Special relativity
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B05 Fourier series and coefficients in several variables
30H10 Hardy spaces

Citations:

Zbl 1143.42006
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Full Text: DOI

References:

[1] Abłamowicz, R.; Fauser, B., On the transposition anti-involution in real Clifford algebras I: the transposition map, J. Linear Multilinear Algebra, 59, 12, 1331-1358 (2011) · Zbl 1393.15029 · doi:10.1080/03081087.2010.517201
[2] Buchholz, S., Tachibana, K., Hitzer, E.: Optimal learning rates for Clifford neurons. In: de Sá J.M., Alexandre L.A., Duch W., Mandic D. (eds) Artificial Neural Networks—ICANN 2007. Lecture Notes in Computer Science, vol. 4668. Springer, Berlin, pp 864-873 (2007). 10.1007/978-3-540-74690-4_88
[3] Buchholz, S., Hitzer, E., Tachibana, K.: Coordinate independent update formulas for versor Clifford neurons. In: Proc. Joint 4th Int. Conf. on Soft Comp. and Intel. Sys., and 9th Int. Symp. on Adv. Intel. Sys., 17-21 Sep. 2008, Nagoya, Japan, pp. 814-819 (2008). 10.14864/softscis.2008.0.814.0
[4] Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. thesis, Christian-Albrechts-Universität zu Kiel (1999) · Zbl 0932.68122
[5] Clifford, WK, Applications of Grassmann’s extensive algebra, Am. J. Math. Pure Appl., 1, 350-358 (1878) · JFM 10.0297.02
[6] Doran, C.; Lasenby, A., Geometric Algebra for Physicists (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1078.53001
[7] El Haoui, Y.; Fahlaoui, S., The uncertainty principle for the two-sided quaternion Fourier transform, Mediterr. J. Math., 14, 221 (2017) · Zbl 1382.42006 · doi:10.1007/s00009-017-1024-5
[8] El Haoui, Y.; Fahlaoui, S., Donoho-Stark’s uncertainty principles in real Clifford algebras, Adv. Appl. Clifford Algebras, 29, 94 (2019) · Zbl 1428.43004 · doi:10.1007/s00006-019-1015-7
[9] El Haoui, Y.; Fahlaoui, S., Miyachi’s theorem for the quaternion Fourier transform, Circuits Syst. Signal Process., 39, 2193-2206 (2020) · Zbl 1508.42011 · doi:10.1007/s00034-019-01243-6
[10] Hitzer, E.: Relativistic physics as application of geometric algebra. In: Adhav, K. (ed.) Proc. of the Int. Conf. on Relativity 2005 (ICR2005). University of Amravati, India, pp. 71-90 (2005). See also: arXiv:1306.0121
[11] Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras. 17, 497-517 (2007). 10.1007/s00006-007-0037-8. See also: arXiv:1306.1023 · Zbl 1143.42006
[12] Hitzer, E.; Mawardi, B., Clifford Fourier transform on multivector fields and uncertainty principles for dimensions \(n = 2\)(mod 4) and \(n = 3\)(mod 4), Adv. Appl. Clifford Algebras, 18, 715-736 (2008) · Zbl 1177.15029 · doi:10.1007/s00006-008-0098-3
[13] Hitzer, E., Directional uncertainty principle for quaternion Fourier transform, Adv. Appl. Clifford Algebras, 20, 271-284 (2010) · Zbl 1198.42006 · doi:10.1007/s00006-009-0175-2
[14] Hitzer, E.: Creative Peace License. http://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/
[15] Hitzer, E., Helmstetter, J., Abłamowicz, R.: Square roots of \(-1\) in real Clifford algebras. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM), vol. 27. Birkhäuser, pp. 123-153 (2013). 10.1007/978-3-0348-0603-9_7. See also: arXiv:1204.4576 · Zbl 1273.15025
[16] Hitzer, E.: The Clifford Fourier transform in real Clifford algebras. In: Hitzer, E., Tachibana, K. (eds.) Session on Geometric Algebra and Applications, IKM 2012, Special Issue of Clifford Analysis, Clifford Algebras and their Applications, vol. 2(3), pp. 227-240 (2013). See also: arXiv:1306.0130 · Zbl 1297.43006
[17] Hitzer, E.; Sangwine, SJ, Quaternion and Clifford Fourier transforms and wavelets, Trends in Mathematics (2013), Basel: Birkhäuser, Basel
[18] Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s Geometric Algebra, Adv. Appl. Clifford Alg., vol. 23, Online First, March 2013, pp. 377-404 (2013). 10.1007/s00006-013-0378-4. See also: arXiv:1305.5663 · Zbl 1269.15022
[19] Hitzer, E., Two-sided Clifford Fourier transform with two square roots of \(-1\) in \(Cl(p, q)\), Adv. Appl. Clifford Algebras, 24, 313-332 (2014) · Zbl 1301.42017 · doi:10.1007/s00006-014-0441-9
[20] Hitzer, E., General steerable two-sided Clifford Fourier transform, convolution and Mustard convolution, Adv. Appl. Clifford Algebras, 27, 3, 2215-2234 (2017) · Zbl 1382.42007 · doi:10.1007/s00006-016-0687-5
[21] Hitzer, E., Special relativistic Fourier transformation and convolutions, Math. Methods Appl. Sci., 42, 2244-2255 (2019) · Zbl 1414.42008 · doi:10.1002/mma.5502
[22] Jday, R., Heisenberg’s and Hardy’s uncertainty principles in real Clifford algebras, Integr. Transforms Spec. Funct., 29, 8, 663-677 (2018) · Zbl 1457.43002 · doi:10.1080/10652469.2018.1483363
[23] Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Course of Theoretical Physics, 4th edn [1939]. Butterworth-Heinemann. ISBN 0 7506 2768 9 (2002)
[24] Linares, F.; Ponce, G., Introduction to Nonlinear Dispersive Equations (2004), Rio de Janeiro: Publicações Matemáticas, IMPA, Rio de Janeiro
[25] Mawardi, B.; Hitzer, E., Clifford Fourier transform and uncertainty principle for the Clifford geometric algebra \(Cl(3,0)\), Adv. Appl. Clifford Algebras, 16, 1, 41-61 (2006) · Zbl 1133.42305 · doi:10.1007/s00006-006-0003-x
[26] Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991) (first published) · Zbl 0733.43001
[27] Laville, G., Ramadanoff, I.P.: Stone-Weierstrass Theorem. See also: arXiv:math/0411090
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