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Approximation of linear canonical wavelet transform on the generalized Sobolev spaces

  • Akhilesh PrasadEmail author
  • Z. A. Ansari
Article
  • 13 Downloads

Abstract

The main objective of this paper is to study the linear canonical wavelet transform (LCWT) on generalized Sobolev space \(B^{\xi ,A}_{p,k}(\mathbb {R})\) and generalized weighted space \(L^{s,p}_{\epsilon ,A}(\mathbb {R})\). Its approximation properties and convergence of convolution for \(F^{A}_{\psi }\) in the space \(B^{\xi ,A}_{p,k}(\mathbb {R})\) are also discussed. Based on these properties, we prove that the LCWT is linear continuous mapping on the spaces of \(F^{*}_{p,A}\) and \( U^{k}_{p,A}\). The composition of LCWTs is defined and studied some results related to it. Moreover, the boundedness results of LCWT as well as composition of LCWTs on the space \(H^{s}_{\epsilon ,A}(\mathbb {R})\) are studied.

Keywords

Linear canonical transform Linear canonical wavelet transform Canonical convolution Generalized Sobolev spaces Schwartz space Generalized weighted Sobolev space 

Mathematics Subject Classification

43A32 42C40 46E35 46F12 

Notes

References

  1. 1.
    Abe, S., Sheridan, J.T.: Optical operations on wave functions as the abelian subgroups of the special affine Fourier transformations. Opt. Lett. 19(22), 1801–1803 (1994)CrossRefGoogle Scholar
  2. 2.
    Abe, S., Sheridan, J.T.: Almost Fourier and almost Fresnel transformation. Opt. Commun. 133, 385–388 (1995)CrossRefGoogle Scholar
  3. 3.
    Alieva, T., Bastiaans, M.J.: Properties of the linear canonical integral transformation. J. Opt. Soc. Am. A 24, 3658–3665 (2007)CrossRefGoogle Scholar
  4. 4.
    Almeida, L.B.: The fractional Fourier transform and time frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)CrossRefGoogle Scholar
  5. 5.
    Bastiaans, M.J.: Propagation laws for the second order moments of the Wigner distribution function in first order optical systems. Optik 82, 173–181 (1989)Google Scholar
  6. 6.
    Bernardo, L.M.: ABCD matrix formalism of fractional Fourier optics. Opt. Eng. 35(3), 732–740 (1996)CrossRefGoogle Scholar
  7. 7.
    Bjorck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6, 351–407 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, L., Zhao, D.: Optical image encryption based fractional wavelet transform. Opt. Commun. 254, 361–367 (2005)CrossRefGoogle Scholar
  9. 9.
    Chui, C.K.: An Introduction to Wavelets. Academic Press, New York (1992)zbMATHGoogle Scholar
  10. 10.
    Chui, C.K.: Wavelet: A mathematical tool for signal analysis. SIAM Publication, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)CrossRefGoogle Scholar
  12. 12.
    Daubechies, I.: Ten lectures on wavelets. In: CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM Publication, Philadelphia (2006)Google Scholar
  13. 13.
    Debnath, L.: Wavelet transforms and their applications. Birkhäuser, Boston (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, Y., Li, B.Z.: The linear canonical wavelet transform on some function spaces. Int. J. Wavelets Multiresolut. Inf. Process. 16, 1850010 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Healy, J.J., Kutay, M.M., Ozaktas, H.M., Sheridan, J.T.: Linear Canonical Transform: Theory and Applications, vol. 198. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  16. 16.
    Herson, D.L.J., Heywood, P.: On the range of some fractional integrals. J. Lond. Math. Soc. 8(4), 607–614 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kou, K., Xu, R.: Windowed linear canonical transform and its applications. Signal Process. 92, 179–188 (2012)CrossRefGoogle Scholar
  18. 18.
    Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representation. J. Math. Phys. 12, 1772–1783 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ozaktas, H.M., Zelevsky, Z., Kutay, M.A.: Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, Hoboken (2001)Google Scholar
  21. 21.
    Pathak, R.S.: The Wavelet Transform, vol. 6. Atlantis Press/Word Scientific, Amsterdam-Paris (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Pei, S.C., Ding, J.J.: Relation between fractional operations and time-frequency distributions and their applications. IEEE Trans. Signal Process. 49, 1638–1655 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pei, S.C., Ding, J.J.: Eigen functions of the canonical transform and self-imaging problems in optical system. In: Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, pp. 73–76. Istanbul, Turkey (2000)Google Scholar
  24. 24.
    Perrier, V., Basdevant, C.: Besov norms in terms of the continuous wavelet transform. Application to structure functions. Math. Models Methods Appl. Sci. 6(5), 649–664 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Prasad, A., Ansari, Z.A.: Continuous wavelet transform involving linear canonical transform. Natl. Acad. Sci. Lett. (2018).  https://doi.org/10.1007/s40009-018-0743-x
  26. 26.
    Prasad, A., Kumar, P.: Composition of continuous fractional wavelet transforms. Natl. Acad. Sci. Lett. 39(2), 115–120 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Prasad, A., Kumar, P.: Fractional continuous wavelet transform on some function spaces. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 86(1), 57–64 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Prasad, A., Kumar, P.: The continuous fractional wavelet transform on generalized weighted Sobolev spaces. Asian-Eur. J. Math. 8(3), 1550054 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rieder, A.: The wavelet transform on Sobolev spaces and its approximation properties. Numer. Math. 58(8), 875–894 (1991)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wolf, K.B.: Integral Transform in Science and Engineering in Canonical Transform, Ch. 9. Plenum, New York (1979)CrossRefGoogle Scholar
  31. 31.
    Zhang, Z.C.: New convolution and product theorem for the linear canonical transform and its applications. Optik 127, 4894–4902 (2016)CrossRefGoogle Scholar
  32. 32.
    Zhang, Z.C.: Unified Wigner-ville distribution and ambiguity function in the linear canonical transform domain. Signal Process. 114, 45–60 (2015)CrossRefGoogle Scholar
  33. 33.
    Zhang, Z.C.: New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain. Signal Process. 118, 51–61 (2016)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Applied Mathematics, Indian Institute of TechnologyIndian School of MinesDhanbadIndia

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