Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T14:28:32.264Z Has data issue: false hasContentIssue false
  • English
  • Français

A distributional Hardy transformation

Published online by Cambridge University Press:  24 October 2008

R. S. Pathak  and
J. N. Pandey
R. S. Pathak
Affiliation:
Department of Mathematics, Banaras Hindu University, Varanasi (U.P.), India and Carleton University, Ottawa, Ontario, Canada
J. N. Pandey
Affiliation:
Department of Mathematics, Banaras Hindu University, Varanasi (U.P.), India and Carleton University, Ottawa, Ontario, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper the classical Hardy transformation (5) has been extended to distributions. The celebrated Hankel transformation, Y-transformation and H-transformation are particular cases of the Hardy transformation. The Hankel transformation has been extended to distributions by Zemanian(12). The distributional theory has been explored by Zemanian(13), Koh and Zemanian(6), and Dubey and Pandey(2). The famous Y- and its reciprocal H-transformations have not been extended to distributions. The inversion and some of the properties of these distributional transformations can be deduced as particular cases of the results contained in this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 247 - 262
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cooke, R. G.The inversion formulae of Hardy and Titchmarsh. Proc. London. Math. Soc. 24 (1925), 381–420.Google Scholar
(2)Dubey, L. S. and Pandey, J. N.On the Hankel transformation of a class of generalized functions (Carleton Maths. Series, 1972).Google Scholar
(3)Erdélyi, A. (Editor). Tables of integral transforms, vol. II (McGraw-Hill Book Co., Inc.; New York, 1954).Google Scholar
(4)Gray, A., Mathews, G. B. and MacRobert, T. M.A treatise on Bessel functions and their applications to physics (McMillan and Co. Ltd; London, Second edn., 1952).Google Scholar
(5)HARDY, G. H.Some formulae in the theory of Bessel functions. Proc. London Math. Soc. 23 (1925), ix.Google Scholar
(6)Koh, E. and Zemanian, A. H.The complex Hankel and I transformations of generalized functions. SIAM J. Appl. Math. 16 (1968), 945–957.CrossRefGoogle Scholar
(7)Magnus, W., Oberhettinger, F. and Soni, R. P.Formulas and theorems for the special functions of mathematical physics (Springer Verlag; New York, 1966).CrossRefGoogle Scholar
(8)Pandey, J. N. and Zemanian, A. H.Complex inversion for the generalized convolution transformation. Pacific J. Math. 25 (1968), 147–157.CrossRefGoogle Scholar
(9)Pandey, J. N.An extension of Haimo's form of Hankel convolutions. Pacific J. Math. 28 (1969), 641–651.CrossRefGoogle Scholar
(10)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford, Second edn., 1962).Google Scholar
(11)Watson, G. N.A treatise on the theory of Bessel functions (Cambridge University Press, Second edn., 1962).Google Scholar
(12)Zemanian, A. H.A distributional Hankel transformation. SIAM J. Appl. Math. 14 (1966), 561–576.CrossRefGoogle Scholar
(13)Zemanian, A. H.Hankel transforms of arbitrary order. Duke Math. J. 34 (1967), 761–769.CrossRefGoogle Scholar
(14)Zemanian, A. H.Generalized integral transformations, pp, 561–771. (Interscience Publishers, 1968).Google Scholar