Two versions of fractional powers of Hankel-type transformations and pseudo-differential operators

  • Akhilesh Prasad
  • Kanailal Mahato


Two versions of the fractional powers of Hankel-type transformations are discussed on certain Zemanian type spaces. The operational formulae are developed. Pseudo-differential operators (p.d.o.) associated with the symbol a(xy) are defined. Integral representation of p.d.o. are obtained. Using the fractional Hankel convolution it is shown that the p.d.o. satisfy the \(L_{\nu ,\mu }^{1}\) norm inequality. Finally these Hankel-type transformations are used in the solution of some partial differential equations.


Hankel transformation Pseudo-differential operator  Bessel type operator Zemanian space 

Mathematics Subject Classification

Primary 46F05 47G30 Secondary 35S05 46E10 46E35 



This work is supported by Indian School of Mines, Dhanbad, under Grant Number No. 613002/ISM JRF/Acad/2013-14.


  1. 1.
    Bateman, H.: Tables of integral transforms, vol. II. McGraw-Hill Book Company, Inc, New York (1954)Google Scholar
  2. 2.
    Betancor, J.J.: A convolution operation for a distributional Hankel tarnsformation. Stud. Math. 117(1), 57–72 (1992)MathSciNetGoogle Scholar
  3. 3.
    Hamio, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kerr, F.H.: Fractional powers of Hankel transforms in the Zemanian space. J. Math. Anal. Appl. 166, 65–83 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Koh, E.L.: The n-dimensional distributional Hankel transformation. Can. J. Math. 27(2), 423–433 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Koh, E.L., Zemanian, A.H.: The complex Hankel and I-transformation of generalized functions. SIAM J. Appl. Math. 16(5), 945–957 (1968)CrossRefzbMATHGoogle Scholar
  7. 7.
    Linares, M., Prez Mendez, J.M.R.: A Hankel type integral transformation on certain space of distributions. Bull. Cal. Math. Soc. 83, 447–546 (1991)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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Namias, V.: Fractionalization of Hankel transforms. J. Inst. Math. Appl. 26, 187–197 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pathak, R.S.: Integral transforms of generalized function and their applications. Gordon Breach Science Publishers, Amsterdem (1997)zbMATHGoogle Scholar
  11. 11.
    Pathak, R.S., Pandey, P.K.: A class of pseudo-differential operators associated with Bessel operators. J. Math. Anal. Appl. 196, 736–747 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Prasad, A., Mahato, A.: The continuous fractional Bessel wavelet transformation. Bound. Value. Probl. 2013(1), 1–16 (2013)Google Scholar
  13. 13.
    Prasad, A., Kumar, M.: Product of two generalized pseudo-differential operators involving fractional Fourier transform. J. Pseudo Diff. Oper. Appl. 2, 355–365 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sheppard, C.J.R., Larkin, K.G.: Similarity theorems for fractional Fourier transforms and fractional Hankel transforms. Opt. Commun. 154, 173–178 (1998)CrossRefGoogle Scholar
  15. 15.
    Torre, A.: Hankel-type integral transforms and their fractionalization: a note. Integral Transforms Spec. Funct. 19(4), 277–292 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge (1958)Google Scholar
  17. 17.
    Wong, M.W.: An introduction to pseudo-differential operators. World Scientific, Singapore (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zemanian, A.H.: Generalized integral transformations. Interscience Publishers, New York (1968)zbMATHGoogle Scholar
  19. 19.
    Zhang, Y., Funaba, T., Tanno, N.: Self-fractional Hankel functions and their properties. Opt. Commun. 176, 71–75 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian School of MinesDhanbadIndia

Personalised recommendations