Skip to main content
SpringerLink
Log in

Octonion Fourier transform of Lipschitz real-valued functions of three variables on the octonion algebra

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

In this paper, we examine the order of magnitude of the octonion Fourier transform (OFT) for real-valued functions of three variables and satisfiying certain Lipschitz conditions. In addition, using the analog of the operator Steklov, we construct the generalized modulus of smoothness, and also using the Laplacian operator we define the K-functional. We use the octonion Fourier transform (OFT) of real-valued functions of three variables to prove the equivalence between K-functionals and modulus of smoothness in the space of square-integrable functions (in Lebesgue sense).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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. Belkina, E.S., Platonov, S.S.: Equivalence of K-functionnals andmodulus of smoothness constructed by generalized dunkl translations. Izv. Vyssh. Uchebn. Zaved. Mat. 8, 3–15 (2008)

    Google Scholar 

  3. Blaszczyk, L.: A generalization of the octonion fourier transform to 3-D octonion-valued signals - properties and possible applications to 3-D LTI partial differential systems. Multidimensional Systems and Signal Processing (2020)

  4. Blaszczyk, L., Snopek, K.M.: Octonion fourier transform of real-valued functions of three variables - selected properties and examples. Signal Process. 136, 29–37 (2017)

    Article  Google Scholar 

  5. Bouhlal, A., Achak, A., Daher, R., Safouane, N.: Dini-Lipschitz functions for the quaternion linear canonical transform. Rendiconti del Circolo Matematico di Palermo Series 2, (2020)

  6. 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 

  7. Bray, W.O.: Growth and Integrability of Fourier Transforms on Euclidean Space. J. Fourier Anal Appl (2014). https://doi.org/10.1007/s00041-014-9354-1

  8. Chanyal, B.C., Bisht, P.S., Negi, O.P.S.: Generalized octonion electrodynamics. Int. J. Theor. Phys. 49, 1333–1343 (2010)

    Article  MathSciNet  Google Scholar 

  9. Dai, Feng: Some equivalence theorems with K-Functionals. J. Appr. Theory 121, 143–157 (2003)

    Article  MathSciNet  Google Scholar 

  10. Dickson, L.E.: On quaternions and their generalization and the history of the eight square theorem. Ann. Math. 20(3), 155–171 (1919)

    Article  MathSciNet  Google Scholar 

  11. Ditzian, Z., Totik, V.: Moduli of smoothness, moduli of smoothness. Springer-Verlag, New York etc. (1987)

    Book  Google Scholar 

  12. 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 

  13. Hahn, S.L., Snopek, K.M.: The unified theory of n-dimensional complex and hypercomplex analytic signals. Bull. Polish Ac. Sci., Tech. Sci. 59(2), 167–181 (2011)

    MATH  Google Scholar 

  14. Kaplan, A.: Quaternions and octonions in mechanics. Rev. De. la Unión Mathe. Argent. 49(2), 45–53 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Peetre, J.: A theory of interpolation of normed spaces, notes de Universidade de Brasilia, (1963)

  16. Potapov, M.K.: Application of the operator of generalized translation in approximation theory. Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika 3, 38–48 (1998)

    MATH  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. Cambridge Phil. Soc. 85, 199–225 (1979)

  19. Snopek, K.M.: New hypercomplex analytic signals and fourier transforms in cayleydickson algebras. Electron. Telecommun. Quater. 55(3), 403–415 (2009)

    Google Scholar 

  20. Snopek, K. M.: The n-d analytic signals and fourier spectra in complex and hypercomplex domains, In: Proc. 34th Int. Conf. on Telecommunications and Signal Processing, Budapest, pages 423-427, (2011)

  21. Snopek, K.M.: The study of properties of n-d analytic signals in complex and hypercomplex domains. Radioengineering 21(2), 29–36 (2012)

    Google Scholar 

  22. Snopek, K.M.: Quaternions and octonions in signal processing-fundamentals and some new results, telecommunication review + telecommunication news. Tele- Radio-Electron. Inform. Technol. 6, 618–622 (2015)

    Google Scholar 

  23. Titchmarsh, E.C.: Introduction to the theory of Fourier integrals, pp. 115–118. Clarendon Press, Oxford (1937)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  26. Younis, M.S.: Fourier Transforms of Lipschitz Functions on Certain Lie Groups. Internat. J. Math. and Math. Sci. 439–448, (2001)

  27. Younis, M.S.: Fourier transforms of Lipschitz functions on compact groups, Ph. D. Thesis. McMaster University (Hamilton, Ont., Canada, 1974)

  28. Younis, M.S.: The Fourier transforms of Lipschitz functions on the heisenberg groups. Internat. J. Math. and Math. Sci. 24(1), 5–9 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. Bouhlal

  2. Labo DGTIC, Department of Mathematics, CRMEF El Jadida, El Jadida, Morocco

    J. Igbida

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

    N. Safouane

Authors
  1. A. Bouhlal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Igbida
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Safouane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bouhlal.

Additional information

I would like It to be in memory of Aicha Boudari ‘Always on our mind, forever in our heart’

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouhlal, A., Igbida, J. & Safouane, N. Octonion Fourier transform of Lipschitz real-valued functions of three variables on the octonion algebra. J. Pseudo-Differ. Oper. Appl. 12, 27 (2021). https://doi.org/10.1007/s11868-021-00405-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11868-021-00405-y

Keywords

Mathematics Subject Classification

Search

Navigation