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Wavelet Transform of Dini Lipschitz Functions on the Quaternion Algebra

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Abstract

In this present work, we generalize Titchmarsh’s theorem for the complex- or hypercomplex-valued functions. Firstly, we examine the order of magnitude of the windowed linear canonical transform (WLCT) of complex-valued functions that achieved certain Lipschitz conditions on \({\mathbb {R}}\). Secondly, we studied the order of magnitude of the 2-D continuous quaternion wavelet transform (CQWT) of certain quaternionic valued Lipschitz functions.

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References

  1. Achak, A., Bouhlal, A., Daher, R., Safouane, N.: Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform. Boletín de la Sociedad Matemática Mexicana (2020)

  2. Ahadi, A., Khoshnevis, A., Saghir, M.Z.: Windowed Fourier transform as an essential digital interferometry tool to study coupled heat and mass transfer. Opt. Laser Technol. 57, 304–317 (2014)

    Article  ADS  Google Scholar 

  3. Bouhlal, A., Achak, A., Daher, R., Safouane, N.: Dini–Lipschitz functions for the quaternion linear canonical transform. Rend. Circ. Mat. Palermo Ser. 2 (2020)

  4. Bray, W.O., Pinsky, M.A.: Growth properties of Fourier transforms via moduli of continuity. J. Funct. Anal. 255, p2265–2285 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bray, W.O.: Growth and integrability of Fourier transforms on Euclidean space. J.Fourier Appl. Anal. (2014). https://doi.org/10.1007/s00041-014-9354-1

  6. Debnath, L., Shankara, B.V., Rao, N.: On new two-dimensional Wigner–Ville nonlinear integral transforms and their basic properties. Integr. Transform. Spec. Funct. 21(3), 165–174 (2010)

    Article  MathSciNet  Google Scholar 

  7. Fahlaoui, S., Boujeddaine, M., El Kassimi, M.: Fourier transforms of Dini–Lipschitz functions on rank 1 symmetric spaces. Mediterr. J. Math. 13(6), 4401–4411 (2016)

    Article  MathSciNet  Google Scholar 

  8. Fan, X.L., Kou, K.I., Liu, M.S.: Quaternion Wigner–Ville distribution associated with the linear canonical transforms. Signal Process. 130, 129–141 (2017)

    Article  Google Scholar 

  9. Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  10. Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 17, 497–517 (2007)

    Article  MathSciNet  Google Scholar 

  11. Hitzer, E.: Directional uncertainty principle for quaternion Fourier transforms. Adv. Appl. Clifford Algebras 20(2), 271–84 (2010)

    Article  MathSciNet  Google Scholar 

  12. Hu, B., Zhou, Y., Lie, L.D., Zhang, J.Y.: Polar linear canonical transform in quaternion domain. J. Inf. Hiding Multimed. Signal Process. 6(6), 1185–1193 (2015)

    Google Scholar 

  13. Kou, K.I., Xu, R.H.: Windowed linear canonical transform and its applications. Signal Process. 92(1), 179–188 (2012)

    Article  Google Scholar 

  14. Mawardi, B., Ashino, R.: Some properties of windowed linear canonical transform and its logarithmic uncertainty principle. Int. J. Wavelets Multiresolution Inf. Process. 14(3) (2016)

  15. Mawardi, B., Ashino, R., Vaillancourt, R.: Two-dimensional quaternion wavelet transform. Appl. Math. Comput. 218, 10–21 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Mawardi, B., Hitzer, E., Hayashi, A., Ashino, R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56, 2411–2417 (2008)

    Article  MathSciNet  Google Scholar 

  17. Platonov, S.S.: An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups, p-adic numbers. Ultrametric Anal. Appl. 9(4), 306–313 (2017)

    Article  MathSciNet  Google Scholar 

  18. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)

    Article  MathSciNet  Google Scholar 

  19. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals, pp. 115–118. Clarendon Press, Oxford (1937)

    MATH  Google Scholar 

  20. Xiang, Q., Qin, K.-Y.: On the relationship between the linear canonical transform and the Fourier transform. In: 2011 4th International Congress on Image and Signal Processing (CISP), pp. 2214–2217

  21. Younis, M.S.: Fourier transforms of Dini–Lipschitz functions. Int. J. Math. Math. Sci. 9(2), 301–312 (1986)

    Article  MathSciNet  Google Scholar 

  22. Younis, M.S.: Fourier transforms on \(L^p\) spaces. Int. J. Math. Math. Sci. 9(2), 301–312 (1986)

    Article  Google Scholar 

  23. Younis, M.S.: Fourier transforms of Lipschitz functions on certain Lie groups. Int. J. Math. Math. Sci., 439–448 (2001)

  24. Younis, M.S.: Fourier Transforms of Lipschitz Functions on Compact Groups, Ph.D. Thesis. McMaster University, Hamilton (1974)

  25. Younis, M.S.: The Fourier transforms of Lipschitz functions on the Heisenberg groups. Int. J. Math. Math. Sci. 24(1), 5–9 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Labo Math Appli, Faculty of Sciences, University Chouaib Doukkali, B.P. 20 , El Jadida, Morocco

    A. Bouhlal

  2. Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca, 20100, Morocco

    N. Safouane & R. Daher

  3. École Supérieure d’Education et de Formation, University Chouaib Doukkali, El Jadida, Morocco

    A. Achak

Authors
  1. A. Bouhlal
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  2. N. Safouane
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  3. A. Achak
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  4. R. Daher
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Corresponding author

Correspondence to N. Safouane.

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Communicated by Eckhard Hitzer

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Bouhlal, A., Safouane, N., Achak, A. et al. Wavelet Transform of Dini Lipschitz Functions on the Quaternion Algebra. Adv. Appl. Clifford Algebras 31, 8 (2021). https://doi.org/10.1007/s00006-020-01112-5

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  • DOI: https://doi.org/10.1007/s00006-020-01112-5

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