Advertisement

A Pair of Linear Canonical Hankel Transformations of Random Order

  • Akhilesh PrasadEmail author
  • Tanuj Kumar
Article
  • 27 Downloads

Abstract

Present paper is devoted to study a pair of linear canonical Hankel transformations of random order and its inverse. Some interesting properties of these transformations are given. Finally, these transformations are used to obtain the solution of some partial differential equations involving Bessel type differential operators.

Keywords

Linear canonical transformation Hankel transformation Zemanian space 

Mathematics Subject Classification

Primary 65R10 46F12 53D22 Secondary 46F05 

Notes

References

  1. 1.
    Bultheel, A., Martínez-Sulbaran, H.: Recent developments in the theory of the fractional Fourier and linear canonical transforms. Bull. Belg. Math. Soc. Simon Stevin 13(5), 971–1005 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)CrossRefGoogle Scholar
  3. 3.
    Erdélyi, A.: Tables of Integral Transformation, vol. 2. McGraw-Hill, New York (1954)Google Scholar
  4. 4.
    Koh, E.L., Li, C.K.: On the inverse of the Hankel transform. Integral Transforms Spec. Funct. 2(4), 279–282 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Koh, E.L., Li, C.K.: The Hankel transformation of Banach-space-valued generalized functions. Proc. Am. Math. Soc. 119(1), 153–163 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Linares, M.L., Mendez Pérez, J.M.R.: Hankel complementary integral transformations of arbitrary order. Int. J. Math. Math. Sci. 15(2), 323–332 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Malgonde, S.P., Bandewar, S.R.: On the generalized Hankel type integral transformation of arbitrary order. Bull. Cal. Math. Soc. 99(3), 251–260 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Malgonde, S.P., Debnath, L.: On Hankel type integral transformations of generalized functions. Integral Transforms Spec. Funct. 15(5), 421–430 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mendez, J.M.: A mixed Parseval equation and the generalized Hankel transformations. Proc. Am. Math. Soc. 102(3), 619–624 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ozaktas, M.H., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)Google Scholar
  11. 11.
    Pathak, R.S.: Integral Transforms of Generalized Functions and Their Applications. Gordon Breach Science Publishers, Amsterdam (1997)zbMATHGoogle Scholar
  12. 12.
    Pei, S.C., Ding, J.J.: Eigenfunctions of linear canonical transform. IEEE Trans. Signal Process. 50(1), 11–26 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pérez, J.M.R.M., Robayna, M.M.S.: A pair of generalized Hankel–Clifford transformations and their applications. J. Math. Anal. Appl. 154(2), 543–557 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Prasad, A., Kumar, P.: Composition of pseudo differential operators associated with fractional Hankel–Clifford integral transformations. Appl. Anal. 95(8), 1792–1807 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Prasad, A., Kumar, T.: A pair of linear canonical Hankel transformations and associated pseudo-differential operators. Appl. Anal. 97(15), 2727–2742 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Prasad, A., Maurya, P.K.: A couple of fractional powers of Hankel-type integral transformations of arbitrary order. Boll. Unione Mat. Ital. 9(3), 323–339 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Torre, A.: Hankel type integral transforms and their fractionalization: a note. Integral Transforms Spec. Funct. 19(4), 277–292 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Torre, A.: Linear and radial canonical transforms of fractional order. J. Comput. Appl. Math. 153(1–2), 477–486 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zemanian, A.H.: Generalized Integral Transforms. Interscience Publishers, New York (1968)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

Personalised recommendations