The uncertainty principle for the two-sided quaternion Fourier transform. (English) Zbl 1382.42006
Summary: In this paper, we provide the Heisenberg’s inequality and the Hardy’s theorem for the two-sided quaternion Fourier transform.
MSC:
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 30G35 | Functions of hypercomplex variables and generalized variables |
References:
| [1] | Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Germany (1999) · Zbl 0932.68122 |
| [2] | Chen, L.P., Kou, K.I., Liu, M.S.: Pitt’s inequality and the uncertainty principle associated with thequaternion Fourier transform. J. Math. Anal. Appl. 423, 681-700 (2015) · Zbl 1425.42012 · doi:10.1016/j.jmaa.2014.10.003 |
| [3] | Christensen, J.G.: Uncertainty Principles. Master’s Thesis, Institute for Mathematical Sciences, University of Copenhagen (2003) · Zbl 1425.42012 |
| [4] | De Bie, H.: New techniques for two-sided quaternion Fourier transform, In: Procedings of AGACSE (2012) · JFM 59.0425.01 |
| [5] | Eckhard, M., Hitzer, S.: Quaternion Fourier transform on quaternion fields and generalizations (2013). arXiv:1306.1023v1 [math.RA] · Zbl 1143.42006 |
| [6] | Ell, T.A.: Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio, pp. 1830-1841 (1993) · Zbl 1369.65190 |
| [7] | Hardy, G.H.: A theorem concerning Fourier transform. J. Lond. Math. Soc. 8, 227-231 (1933) · Zbl 0007.30403 · doi:10.1112/jlms/s1-8.3.227 |
| [8] | Pei, S.C., Ding, J.J., Chang, J.H.: Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), 2783-2797 (2001) · Zbl 1369.65190 · doi:10.1109/78.960426 |
| [9] | Pierce, L.B.: Hardy Functions. Undergraduate Junior Paper , Princeton University, University (2001) · Zbl 1369.65190 |
| [10] | Sitaram, A., Sundari, M.: An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups. Pac. J. Math. 177, 187-200 (1997) · Zbl 0882.43005 · doi:10.2140/pjm.1997.177.187 |
| [11] | Weyl, Gruppentheorie, Quantenmechanik, S., Hirzel, Leipzig (1928). Revised English edition: The Theory of Groups and Qunantum Mechans, Methuen, London (1931); Reprinted by Dover, New York (1950) · Zbl 1338.42013 |
| [12] | Yang, Y., Dang, P., Qian, T.: Tighter uncertainty principles based on quaternion Fourier transform. Adv. Appl. Clifford Algebras 26, 479-497 (2016) · Zbl 1338.42013 · doi:10.1007/s00006-015-0579-0 |
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