Abstract
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(x, y) 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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bateman, H.: Tables of integral transforms, vol. II. McGraw-Hill Book Company, Inc, New York (1954)
Betancor, J.J.: A convolution operation for a distributional Hankel tarnsformation. Stud. Math. 117(1), 57–72 (1992)
Hamio, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)
Kerr, F.H.: Fractional powers of Hankel transforms in the Zemanian space. J. Math. Anal. Appl. 166, 65–83 (1992)
Koh, E.L.: The n-dimensional distributional Hankel transformation. Can. J. Math. 27(2), 423–433 (1975)
Koh, E.L., Zemanian, A.H.: The complex Hankel and I-transformation of generalized functions. SIAM J. Appl. Math. 16(5), 945–957 (1968)
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)
Malgonde, S.P., Debnath, L.: On Hankel type integral transformations of generalized functions. Integral Transforms Spec. Funct. 15(5), 421–430 (2004)
Namias, V.: Fractionalization of Hankel transforms. J. Inst. Math. Appl. 26, 187–197 (1980)
Pathak, R.S.: Integral transforms of generalized function and their applications. Gordon Breach Science Publishers, Amsterdem (1997)
Pathak, R.S., Pandey, P.K.: A class of pseudo-differential operators associated with Bessel operators. J. Math. Anal. Appl. 196, 736–747 (1995)
Prasad, A., Mahato, A.: The continuous fractional Bessel wavelet transformation. Bound. Value. Probl. 2013(1), 1–16 (2013)
Prasad, A., Kumar, M.: Product of two generalized pseudo-differential operators involving fractional Fourier transform. J. Pseudo Diff. Oper. Appl. 2, 355–365 (2011)
Sheppard, C.J.R., Larkin, K.G.: Similarity theorems for fractional Fourier transforms and fractional Hankel transforms. Opt. Commun. 154, 173–178 (1998)
Torre, A.: Hankel-type integral transforms and their fractionalization: a note. Integral Transforms Spec. Funct. 19(4), 277–292 (2008)
Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge (1958)
Wong, M.W.: An introduction to pseudo-differential operators. World Scientific, Singapore (1999)
Zemanian, A.H.: Generalized integral transformations. Interscience Publishers, New York (1968)
Zhang, Y., Funaba, T., Tanno, N.: Self-fractional Hankel functions and their properties. Opt. Commun. 176, 71–75 (2000)
Acknowledgments
This work is supported by Indian School of Mines, Dhanbad, under Grant Number No. 613002/ISM JRF/Acad/2013-14.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Prasad, A., Mahato, K. Two versions of fractional powers of Hankel-type transformations and pseudo-differential operators. Rend. Circ. Mat. Palermo, II. Ser 65, 209–241 (2016). https://doi.org/10.1007/s12215-015-0229-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-015-0229-3