Geodesic
On Hankel Transformable Spaces and a Cauchy Problem
Canadian journal of mathematics, Tome 37 (1985) no. 1, pp. 84-106

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Pathak, R. S. On Hankel Transformable Spaces and a Cauchy Problem. Canadian journal of mathematics, Tome 37 (1985) no. 1, pp. 84-106. doi: 10.4153/CJM-1985-008-2
@article{10_4153_CJM_1985_008_2,
     author = {Pathak, R. S.},
     title = {On {Hankel} {Transformable} {Spaces} and a {Cauchy} {Problem}},
     journal = {Canadian journal of mathematics},
     pages = {84--106},
     year = {1985},
     volume = {37},
     number = {1},
     doi = {10.4153/CJM-1985-008-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1985-008-2/}
}
TY  - JOUR
AU  - Pathak, R. S.
TI  - On Hankel Transformable Spaces and a Cauchy Problem
JO  - Canadian journal of mathematics
PY  - 1985
SP  - 84
EP  - 106
VL  - 37
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1985-008-2/
DO  - 10.4153/CJM-1985-008-2
ID  - 10_4153_CJM_1985_008_2
ER  - 
%0 Journal Article
%A Pathak, R. S.
%T On Hankel Transformable Spaces and a Cauchy Problem
%J Canadian journal of mathematics
%D 1985
%P 84-106
%V 37
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1985-008-2/
%R 10.4153/CJM-1985-008-2
%F 10_4153_CJM_1985_008_2

[1] 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] 2. Dube, L. S. and Pandey, J. N., On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975), 337–354. Google Scholar

[3] 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] 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] 5. Friedman, A., Generalized functions and partial differential equations (Prentice-Hall, New Jersey, 1963). Google Scholar | DOI

[6] 6. Gelfand, I. M. and Shilov, G. E., Generalized functions, vol. II (Academic Press, New York, 1968). Google Scholar

[7] 7. Gelfand, I. M. and Shilov, G. E., Generalized functions, vol. III (Academic Press, New York, 1967). Google Scholar

[8] 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] 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] 10. Koh, E. L., The n-dimensional distributional Hankel transformation, Can. J. Math. 22 (1975), 423–433. Google Scholar

[11] 11. Lee, W. Y. K., On spaces of type H and their Hankel transformations, SIAM J. Math. Anal. 5 (1974), 336–347. Google Scholar

[12] 12. Lee, W. Y. K., On the Cauchy problem of the differential operator S , Proc. Amer. Math. Soc. 51 (1975), 149–154. Google Scholar

[13] 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] 14. Pathak, R. S. and Pandey, A. B., On Hankel transforms of ultradistributions, Communicated. Google Scholar | DOI

[15] 15. Pathak, R. S. and Pandey, J. N., A distributional Hardy transformation, Proc. Camb. Phil. Soc. 76 (1974), 247–262. Google Scholar

[16] 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] 17. Sneddon, I. N., The use of integral transforms (McGraw-Hill Book Co., New York, 1972). Google Scholar

[18] 18. Titchmarsh, E. C., The theory of functions, 2nd edn. (Oxford Univ. Press, 1939). Google Scholar

[19] 19. Treves, F., Topological vector spaces, distributions and kernels (Academic Press, New York, 1967). Google Scholar

[20] 20. Zemanian, A. H., Generalized integral transformations (Interscience Publishers, New York, 1968). Google Scholar

[21] 21. Zemanian, A. H., A distributional Hankel transformation, SIAM J. Appl. Math. 14 (1966), 561–576. Google Scholar

[22] 22. Zemanian, A. H., The Hankel transformation of certain distributions of rapid growth, SIAM J. Appl. Math. 74 (1966), 678–690. Google Scholar

[23] 23. Zemanian, A. H., Hankel transforms of arbitrary order, Duke Math. J. 34 (1967), 761–770. Google Scholar

Cité par Sources :