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On Hankel Transformable Spaces and a Cauchy Problem

Published online by Cambridge University Press:  20 November 2018

R. S. Pathak
R. S. Pathak*
Affiliation:
King Saud University, Riyad, Saudi Arabia
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The classical Hankel transform of a conventional function ϕ on (0, ∞) defined formally by

was extended by Zemanian [21-23] to certain generalized functions of one dimension. Koh [9, 10] extended the work of [21] to n-dimensions, and that of [22] to arbitrary real values of μ. Motivated from the work of Gelfand and Shilov [6], Lee [11] introduced spaces of type Hμ and studied their Hankel transforms. The results of Lee [11] and Zemanian [21] are special cases of recent results obtained by the author and Pandey [14]. The aforesaid extensions are accomplished by using the so-called adjoint method of extending integral transforms to generalized functions. Dube and Pandey [2], Pathak and Pandey [15, 16] applied a more direct method, the so-called kernel method, for extending the Hankel and other related transforms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 1 , 01 February 1985 , pp. 84 - 106
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. de Bruijn, N. G., A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence, Nieu Archief voor Wiskunde 21 (1973), 205–280.Google Scholar
2. Dube, L. S. and Pandey, J. N., On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975), 337–354.Google Scholar
3. van Eijndhoven, S. J. L., A theory of generalized functions based on one parameter groups of unbounded self-adjoint operators, T.H. report 81-WSK-03 (Eindhoven University of Technology, Eindhoven, 1981).Google Scholar
4. van Eijndhoven, S. J. L. and de Graaf, J., Some results on Hankel invariant distribution spaces, Proc. Koninkl. Nederl. Akad. Weten. (Amsterdam), A 86 (1983), 77–87.Google Scholar
5. Friedman, A., Generalized functions and partial differential equations (Prentice-Hall, New Jersey, 1963).CrossRefGoogle Scholar
6. Gelfand, I. M. and Shilov, G. E., Generalized functions, vol. II (Academic Press, New York, 1968).Google Scholar
7. Gelfand, I. M. and Shilov, G. E., Generalized functions, vol. III (Academic Press, New York, 1967).Google Scholar
8. de Graaf, J., A theory of generalized functions based on holomorphic semigroups, TH report 79-WSK-02 (Eindhoven University of Technology, Eindhoven, 1979).Google Scholar
9. Koh, E. L., The Hankel transformation of negative order for distributions of rapid growth, SIAM J. Math. Anal. 1 (1970), 322–327.Google Scholar
10. Koh, E. L., The n-dimensional distributional Hankel transformation, Can. J. Math. 22 (1975), 423–433.Google Scholar
11. Lee, W. Y. K., On spaces of type Hμ and their Hankel transformations, SIAM J. Math. Anal. 5 (1974), 336–347.Google Scholar
12. Lee, W. Y. K., On the Cauchy problem of the differential operator Sμ , Proc. Amer. Math. Soc. 51 (1975), 149–154.Google Scholar
13. McBride, A. C., The Hankel transform of some classes of generalized functions and connections with fractional integration, Proc. Roy. Soc. (Edinburgh) 81 A(1978), 95–117.Google Scholar
14. Pathak, R. S. and Pandey, A. B., On Hankel transforms of ultradistributions, Communicated.CrossRefGoogle Scholar
15. Pathak, R. S. and Pandey, J. N., A distributional Hardy transformation, Proc. Camb. Phil. Soc. 76 (1974), 247–262.Google Scholar
16. Pathak, R. S. and Pandey, J. N., A distributional Hardy transformation II, Internat. J. Math, and Math. Sci. 2 (1976), 693–701.Google Scholar
17. Sneddon, I. N., The use of integral transforms (McGraw-Hill Book Co., New York, 1972).Google Scholar
18. Titchmarsh, E. C., The theory of functions, 2nd edn. (Oxford Univ. Press, 1939).Google Scholar
19. Treves, F., Topological vector spaces, distributions and kernels (Academic Press, New York, 1967).Google Scholar
20. Zemanian, A. H., Generalized integral transformations (Interscience Publishers, New York, 1968).Google Scholar
21. Zemanian, A. H., A distributional Hankel transformation, SIAM J. Appl. Math. 14 (1966), 561–576.Google Scholar
22. Zemanian, A. H., The Hankel transformation of certain distributions of rapid growth, SIAM J. Appl. Math. 74 (1966), 678–690.Google Scholar
23. Zemanian, A. H., Hankel transforms of arbitrary order, Duke Math. J. 34 (1967), 761–770.Google Scholar