SpringerLink
Log in

Novel Quaternionic Fractional Wavelet Transform

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we introduce the notion of a novel quaternionic fractional wavelet transform (FRWT). Firstly, we establish the inversion formula and Parseval theorem for the new integral transform. The necessary and sufficient condition for the \(\alpha \)-order quaternionic FRWT to be the quaternionic FRWT of some function is also given. Towards the end we have given few examples to find the quaternionic FRWT of functions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Ahmad, O., Sheikh, N.A.: Novel special affine wavelet transform and associated uncertainty principles. Int. J. Geom. Methods Mod. Phys. 18(04), 2150055 (2021)

    Article  MathSciNet  Google Scholar 

  2. Ahmad, O., Sheikh, N.A., Shah, F.A.: Fractional biorthogonal wavelets in \(L_2({\mathbb{R}})\). Applicable Analysis, pp.1–22 (2021)

  3. Ahmad, O., Sheikh, N.A., Shah, F.A.: Fractional multiresolution analysis and associated scaling functions in \(L^{2}({\mathbb{R}})\). Anal. Math. Phys. 11(2), 1–20 (2021)

    Article  Google Scholar 

  4. Ahmad, O., Sheikh, N.A., Nisar, K.S., Shah, F.A.: Biorthogonal wavelets on the spectrum. Math. Methods Appl. Sci. 44(6), 4479–4490 (2021)

    Article  MathSciNet  Google Scholar 

  5. Ahmad, O., Achak, A., Sheikh, N.A., Warbhe, U.: Uncertainty principles associated with multi-dimensional linear canonical transform. International Journal of Geometric Methods in Modern Physics, pp. 2250029 (2021)

  6. Ahmad, O., Bhat, M.Y., Sheikh, N.A.: Construction of Parseval Framelets Associated with GMRA on local fields of positive characteristic. Numer. Funct. Anal. Optim. 42(3), 344–370 (2021)

    Article  MathSciNet  Google Scholar 

  7. Ahmad, O., Sheikh, N.A.: Inequalities for wavelet frames with composite dilations in \(L^2({\mathbb{R}}^n)\). Rocky Mountain J. Math. 51(1), 31–41 (2021)

    Article  MathSciNet  Google Scholar 

  8. Ahmad, O., Ahmad, N.: Construction of nonuniform wavelet frames on non-Archimedean fields. Math. Phys. Anal. Geom. 23(4), 1–20 (2020)

    Article  MathSciNet  Google Scholar 

  9. Ali, K.K., Abd El Salam, M.A., Mohamed, E.M., Samet, B., Kumar, S., Osman, M.S.: Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Adv. Difference Equ. 2020(1), 1–23 (2020)

    Article  MathSciNet  Google Scholar 

  10. Arqub, O.A., Osman, M.S., Abdel-Aty, A.H., Mohamed, A.B.A., Momani, S.: A numerical algorithm for the solutions of ABC singular Lane-Emden type models arising in astrophysics using reproducing kernel discretization method. Mathematics 8(6), 923 (2020)

    Article  Google Scholar 

  11. Arqub, O.A., Al-Smadi, M., Almusawa, H., Baleanu, D., Hayat, T., Alhodaly, M., Osman, M.S.: A novel analytical algorithm for generalized fifth-order time-fractional nonlinear evolution equations with conformable time derivative arising in shallow water waves. Alex. Eng. J. 61(7), 5753–5769 (2022)

    Article  Google Scholar 

  12. Bayones, F.S., Nisar, K.S., Khan, K.A., Raza, N., Hussien, N.S., Osman, M.S., Abualnaja, K.M.: Magneto-hydrodynamics (MHD) flow analysis with mixed convection moves through a stretching surface. AIP Adv. 11(4), 045001 (2021)

    Article  Google Scholar 

  13. Bracewell, R.N.: Convolution, The Fourier transform and its applications, vol. 2, pp. 24–50. McGraw-Hill, New York (1986)

    Google Scholar 

  14. Capus, C., Brown, K.: Short-time fractional Fourier methods for the time-frequency representation of chirp signals. J. Acoustical Soc. Amer. 113(6), 3253–3263 (2003)

    Article  Google Scholar 

  15. Delsuc, M.A.: Spectral representation of 2D NMR spectra by hypercomplex numbers. J. Magnetic Resonance (1969) 77(1), 119–124 (1988)

    Article  Google Scholar 

  16. Dhawan, S., Machado, J.A.T., Brzeziński, D.W., Osman, M.S.: A chebyshev wavelet collocation method for some types of differential problems. Symmetry 13(4), 536 (2021)

    Article  Google Scholar 

  17. Djennadi, S., Shawagfeh, N., Osman, M.S., Gómez-Aguilar, J.F., Arqub, O.A.: The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique. Phys. Scr. 96(9), 094006 (2021)

    Article  Google Scholar 

  18. Durak, L., Arikan, O.: Short-time Fourier transform: two fundamental properties and an optimal implementation. IEEE Trans. Signal Process. 51(5), 1231–1242 (2003)

    Article  MathSciNet  Google Scholar 

  19. Ell, T.A.: Hypercomplex spectral transformations [PhD thesis]. Minneapolis: University of Minnesota (1992)

  20. Ell, T.A.: December. Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceedings of 32nd IEEE Conference on Decision and Control (pp. 1830-1841). IEEE (1993)

  21. Hamilton, W.R.: On a new species of imaginary quantities, connected with the theory of quaternions. Proc. R. Irish Acad. 1836–1869(2), 424–434 (1840)

    Google Scholar 

  22. Hamilton, W.R.: On quaternions; or on a new system of imaginaries in algebra, letter to John T. Graves (October 1843) (1843)

  23. He, J., Yu, B.: Continuous wavelet transforms on the space \(L_2({\mathbb{R}},H, dx)\). Appl. Math. Lett. 17(1), 111–121 (2004)

    Article  MathSciNet  Google Scholar 

  24. Huang, Y., Suter, B.: September. Fractional wave packet transform. In: 1996 IEEE Digital Signal Processing Workshop Proceedings (pp. 413-415). IEEE (1996)

  25. Inan, B., Osman, M.S., Ak, T., Baleanu, D.: Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations. Math. Methods Appl. Sci. 43(5), 2588–2600 (2020)

    Article  MathSciNet  Google Scholar 

  26. Khalid, A., Rehan, A., Nisar, K.S., Osman, M.S.: Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit. Phys. Scr. 96(10), 104001 (2021)

    Article  Google Scholar 

  27. Mendlovic, D., Zalevsky, Z., Mas, D., García, J., Ferreira, C.: Fractional wavelet transform. Appl. Opt. 36(20), 4801–4806 (1997)

    Article  Google Scholar 

  28. Ozaktas, H.M., Aytür, O.: Fractional fourier domains. Signal Process. 46(1), 119–124 (1995)

    Article  Google Scholar 

  29. Prasad, A., Manna, S., Mahato, A., Singh, V.K.: The generalized continuous wavelet transform associated with the fractional Fourier transform. J. Comput. Appl. Math. 259, 660–671 (2014)

    Article  MathSciNet  Google Scholar 

  30. Sharma, P.B., Prasad, A.: Composition of Quadratic-Phase Fourier Wavelet Transform. Int. J. Appl. Comput. Math. 8(3), 1–14 (2022)

    MathSciNet  MATH  Google Scholar 

  31. Shi, J., Zhang, N., Liu, X.: A novel fractional wavelet transform and its applications. SCIENCE CHINA Inf. Sci. 55(6), 1270–1279 (2012)

    Article  MathSciNet  Google Scholar 

  32. Sommen, F.: Hypercomplex Fourier and Laplace transforms II. Complex Var. Elliptic Equ. 1(2–3), 209–238 (1983)

    MathSciNet  MATH  Google Scholar 

  33. Sommen, F.: Hypercomplex Fourier and Laplace transforms I. Ill. J. Math. 26(2), 332–352 (1982)

    MathSciNet  MATH  Google Scholar 

  34. Zhao, J., Peng, L.: Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations. J. Nat. Geom. 20(1/2), 21–32 (2001)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar, 190 006, Jammu and Kashmir, India

    Tawseef Ahmad Sheikh & Neyaz A. Sheikh

Authors
  1. Tawseef Ahmad Sheikh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neyaz A. Sheikh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Neyaz A. Sheikh.

Ethics declarations

Competing Interests

The authors have not disclosed any competing interests.

Additional information

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

Sheikh, T.A., Sheikh, N.A. Novel Quaternionic Fractional Wavelet Transform. Int. J. Appl. Comput. Math 8, 162 (2022). https://doi.org/10.1007/s40819-022-01364-8

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-022-01364-8

Keywords

Mathematics Subject Classification

Search

Navigation