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Beurling’s theorem for the quaternion Fourier transform

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Abstract

The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling’s theorem, Hardy, Cowling–Price and Gelfand–Shilov theorems, is obtained for the two-sided quaternion Fourier transform.

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References

  1. Bahri, M., Ashino, R., Vaillancourt, R: Convolution theorems for quaternion Fourier transform: properties and applications. Abstr. Appl. Anal. 2013 , Article ID 162769, p. 10 (2013). https://doi.org/10.1155/2013/162769. http://projecteuclid.org/euclid.aaa/1393512106

    Article  MathSciNet  Google Scholar 

  2. Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the widowed Fourier transform. Rev. Mat. Iberoam. 19, 23–55 (2003)

    Article  Google Scholar 

  3. 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)

  4. Chen, L.P., Kou, K.I., Liu, M.S.: Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J. Math. Anal. Appl. 423, 681–700 (2015)

    Article  MathSciNet  Google Scholar 

  5. Eckhard M., Hitzer, S.: Quaternion Fourier transform on quaternion fields and generalizations, arXiv:1306.1023v1 [math.RA] 5 Jun 2013

  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)

  7. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207–238 (1997)

    Article  MathSciNet  Google Scholar 

  8. Gel’fand, I.M., Shilov, G.E.: Generalized functions, 2. Moscow,1958 (Russian). English translation, Academic Press, (1968)

  9. Hardy, G.H.: A theorem concerning Fourier transform. J. Lond. Math. Soc. 8, 227–231 (1933)

    Article  MathSciNet  Google Scholar 

  10. El Haoui, Y., Fahlaoui, S.: The Uncertainty principle for the two-sided quaternion Fourier transform. Mediterr. J. Math. (2017). https://doi.org/10.1007/s00009-017-1024-5

    Article  MathSciNet  MATH  Google Scholar 

  11. Hörmander, L.: A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Math. 2, 237–240 (1991)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Thangavelu, S.: An introduction to the uncertainty principle, Progr. Math. 217 Birkhauser, Boston, (2003)

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Faculty of Sciences, Equipe d’Analyse Harmonique et Probabilités, University Moulay Ismail, BP 11201, Zitoune, Meknes, Morocco

    Youssef El Haoui & Said Fahlaoui

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  1. Youssef El Haoui
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  2. Said Fahlaoui
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Correspondence to Youssef El Haoui.

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El Haoui, Y., Fahlaoui, S. Beurling’s theorem for the quaternion Fourier transform. J. Pseudo-Differ. Oper. Appl. 11, 187–199 (2020). https://doi.org/10.1007/s11868-019-00281-7

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  • DOI: https://doi.org/10.1007/s11868-019-00281-7

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