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Wave packet transform in the framework of Lebedev–Skalskaya transforms

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Abstract

In this paper, we define the wave-packet transform (WPT) involving Lebedev–Skalskaya transform (LS-transform) and establish some norm estimates of LS-wavelet, LS-wavelet transform, and Plancherel’s relation for WPT. Moreover, we obtain the Calderon-type reproducing formula using LS-transform theory and its convolution.

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References

  1. Bracewell, R.N.: The Fourier transform and its applications, vol. 31999. McGraw-Hill, New York (1986)

    Google Scholar 

  2. Cohen, L.: Time-frequency analysis, vol. 778. Prentice hall, New Jersey (1995)

    Google Scholar 

  3. Debnath, L., Shah, F.A.: Lecture notes on wavelet transforms. Birkhäuser, Boston (2017)

    Book  Google Scholar 

  4. Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations. I. General results. J. Math. Phys. 26(10), 2473–2479 (1985)

    Article  MathSciNet  Google Scholar 

  5. Arefijamaal, A.A., Kamyabi-Gol, R.A.: On the square integrability of quasi regular representation on semidirect product groups. J. Geom. Anal. 19(3), 541–552 (2009)

    Article  MathSciNet  Google Scholar 

  6. Pathak, R.S., Pandey, C.P.: Laguerre wavelet transforms. Integr. Transf. Spec. Funct. 20(7), 505–518 (2009)

    Article  MathSciNet  Google Scholar 

  7. Pathak, R.S., Dixit, M.M.: Continuous and discrete Bessel wavelet transforms. J. Comput. Appl. Math. 160(1), 241–250 (2003)

    Article  MathSciNet  Google Scholar 

  8. Prasad, A., Kumar, P.: Composition of the continuous fractional wavelet transforms. Nat. Acad. Sci. Lett. 39(2), 115–120 (2016)

    Article  MathSciNet  Google Scholar 

  9. Prasad, A., Kumar, S.: Wavelet transformation associated with second Hankel-Clifford transformation. Nat. Acad. Sci. Lett. 38(6), 493–496 (2015)

    Article  MathSciNet  Google Scholar 

  10. Prasad, A., Mandal, U.K.: Wavelet transforms associated with the Kontorovich-Lebedev transform. Int. J. Wavelets Multiresolut. Inf. Process. 15(2), 1750011 (2017)

    Article  MathSciNet  Google Scholar 

  11. Prasad, A., Verma, S.K.: Continuous wavelet transform associated with zero-order Mehler-Fock transform and its composition. Int. J. Wavelets Multiresolution Inf. Process. 16(6), 1850050 (2018)

    Article  MathSciNet  Google Scholar 

  12. Prasad, A., Maan, J., Verma, S.K.: Wavelet transforms associated with the index Whittaker transform. Math. Methods Appl. Sci. 44(13), 10734–10752 (2021)

    Article  MathSciNet  Google Scholar 

  13. Upadhyay, S.K., Mishra, K.K.: The continuous fractional Bessel wavelet transform and its applications. Integr. Transfor. Spec. Funct. (2023). https://doi.org/10.1080/10652469.2023.2164889

    Article  Google Scholar 

  14. Gupt, A.K., Mandal, U.K., Prasad, A.: Lebedev-Skalskaya transform related continuous wavelet transform. (Communicated)

  15. Pathak, A., Pandey, S.: Besov-type spaces associated with Lebedev-Skalskaya wavelet transform. Math. Methods Appl. Sci. 46(14), 15626–15640 (2023)

    Article  MathSciNet  Google Scholar 

  16. Christensen, O., Rahimi, A.: Frame properties of wave packet systems in \(L^2({{\mathbb{R} }}^d)\). Adv. Comput. Math. 29(2), 101–111 (2008)

    Article  MathSciNet  Google Scholar 

  17. Posch, T.E.: The wave packet transform (WPT) as applied to signal processing. In: Proceedings of IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, pp. 143–146. Victoria, BC, Canada (1992)

  18. Farashahi, A.G.: Multivariate wave-packet transforms. Z. Anal. Anwend. 36(4), 481–500 (2017)

    Article  MathSciNet  Google Scholar 

  19. Farashahi, A.G.: Square-integrability of multivariate metaplectic wave-packet representations. J. Phys. A 50(11), 115202 (2017)

    Article  MathSciNet  Google Scholar 

  20. Farashahi, A.G.: Square-integrability of metaplectic wave-packet representations on \(L^2({{\mathbb{R} }})\). J. Math. Anal. Appl. 449(1), 769–792 (2017)

    Article  MathSciNet  Google Scholar 

  21. Farashahi, A.G.: Abstract harmonic analysis of wave packet transforms over locally compact abelian groups. Banach J. Math. Anal. 11(1), 50–71 (2017)

    Article  MathSciNet  Google Scholar 

  22. Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Commun. Partial. Differ. Equ. 3(11), 979–1005 (1978)

    Article  MathSciNet  Google Scholar 

  23. Shah, F.A., Ahmad, O., Jorgensen, P.E.: Fractional wave packet systems in \(L^{ 2}({{\mathbb{R} }})\). J. Math. Phys. 59(7), 073509 (2018)

    Article  MathSciNet  Google Scholar 

  24. Sui, X., Wan, K., Zhang, Y.: Pattern recognition of SEMG based on wavelet packet transform and improved SVM. Optik 176, 228–235 (2019)

    Article  Google Scholar 

  25. Prasad, A., Singh, M.K., Kumar, M.: The continuous fractional wave packet transform. AIP Conf. Proc. 1558(1), 856–859 (2013)

    Article  Google Scholar 

  26. Mandal, U.K., Verma, S.K., Prasad, A.: Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform. Filomat 35(1), 47–60 (2021)

    Article  MathSciNet  Google Scholar 

  27. Mandal, U.K., Prasad, A.: Lebedev-Skalskaya transforms on certain function spaces and associated pseudo-differential operators. Integr. Transforms Spec. Funct. 32(2), 113–133 (2021)

    Article  MathSciNet  Google Scholar 

  28. Mandal, U.K., Prasad, A.: Lebedev-Skalskaya transforms and allied operators on certain function spaces. Integr. Transforms Spec. Funct. 33(4), 320–340 (2022)

    Article  MathSciNet  Google Scholar 

  29. Mandal, U.K., Prasad, A., Gupt, A.K.: Reverse convolution inequalities for Lebedev-Skalskaya transforms. Forum Math. 34(4), 1095–1107 (2022)

    MathSciNet  Google Scholar 

  30. Lebedev, N.N., Skalskaya, I.P.: Some integral transforms related to the Kontorovich-Lebedev transforms. In: The Questions of Mathematical Physics, Leningrad, Nauka (1976), pp. 68-79 (in Russian)

  31. Yakubovich, S.B.: Index transforms (with foreword by H. M. Srivastava), World Scientific Publishing Company, Singapore (1996)

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Acknowledgements

Authors are very thankful to the reviewers for their constructive comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, India

    Ajay K. Gupt & Akhilesh Prasad

  2. Department of Mathematics, Nalanda College, Patliputra University, Biharsharif, Patna, India

    U. K. Mandal

Authors
  1. Ajay K. Gupt
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  2. Akhilesh Prasad
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  3. U. K. Mandal
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Correspondence to Akhilesh Prasad.

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This work is supported by the Council of Scientific and Industrial Research (CSIR), India under file no. 09/085(0123)/2019-EMR-I.

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Gupt, A.K., Prasad, A. & Mandal, U.K. Wave packet transform in the framework of Lebedev–Skalskaya transforms. Acta Sci. Math. (Szeged) 90, 73–89 (2024). https://doi.org/10.1007/s44146-023-00097-4

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