A Mixed Parseval's Equation And a Generalized Hankel Transformation of Distributions

Betancor, J. J.

Canadian journal of mathematics, Tome 41 (1989) no. 2, pp. 274-284

Résumé

Let an integral transform T{f} of a complex valued function f(x) defined over the interval (0, ∞) be defined as One of the most usual procedures to extend the classical transform (l.a) to generalized functions consists in constructing a space A of testing functions over (0, ∞) which is closed with respect to the classical transform (l.a) and then the corresponding transform of the generalized function/ of the dual space of Ais defined through This approach has been followed by L. Schwartz [13] and A. H. Zemanian [20], amongst others.

Détail

DOI : 10.4153/CJM-1989-014-9

Bibliographie

[1] 1. Altenburg, G., Bessel transformationen in Raumen von Grundfunktionen uber dem intervall fi = (0, ∞) und derem dual raumen, Math. Nachr. 108 (1982), 197–218. Google Scholar

[2] 2. Betancor, J.J., La transformation integral /F, Doctoral Thesis, Depart. Math. Anal., Univ. of La Laguna, Tenerife, Spain (1987). Google Scholar

[3] 3. Dimovski, I., Operational calculus for a class of differential operators, Compt. Rend. L’Academie Bulgare Sci. 19 (1966), 1111–1114. Google Scholar

[4] 4. Gelfand, I.M. and Shilov, G.E., Generalized functions, I and 2 (Academic Press, New York, 1964, 1968). Google Scholar

[5] 5. Gray, A., Mathews, G.B. and McRobert, T.M., A treatise on Bessel functions and their applications to physics (McMillan and Co. Ltd., London, 1952). Google Scholar

[6] 6. Koh, E.L. and Zemanian, A.H., The complex Hankel and I-transformations of generalized functions. SIAM J. Appl. Math. 16 (1968), 945–957. Google Scholar

[7] 7. Lee, W.Y., On Schwartz's Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal., 6 (2) (1975), 427–432. Google Scholar

[8] 8. McBride, A., The Hankel transformation of some classes of generalized functions and connections with fractional integration, Proc. Roy. Soc. Edinburgh 81 A (1978), 95–117. Google Scholar

[9] 9. Mendez, J.M., La transformation integral de Hankel-Clifford, Secretariado de Publicaciones de la Universidad de La Laguna, Coleccion Monografias, 8 (Tenerife, Spain, 1981). Google Scholar

[10] 10. Mendez, J.M., On the Bessel transform of arbitrary order, Math. Nachr. 136 (1988), 233–239. Google Scholar

[11] 11. Mendez, J.M., A mixed Parseval equation and the generalized Hankel transformation, Proc. Amer. Math. Soc. 102 (1988), 619–624. Google Scholar

[12] 12. Schuitman, A., On a certain test function space for the Schwartz's Hankel transformation, Delft Progr. Report 2 (1977), 193–206. Google Scholar

[13] 13. Schwartz, L., Théorie des distributions (Hermann, Paris, 1966). Google Scholar

[14] 14. Schwartz, A.L., An inversion theorem for Hankel transforms, Proc. Amer. Math. Soc. 22 (1969), 713–717. Google Scholar

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

[16] 16. Socas, M.M., La transformation integral de Hankel-Clifford en tiertos espacios de funciones generalizadas, Doctoral Thesis, Depart. Math. Anal., Univ. of La Laguna, Tenerife, Spain, 1986. Google Scholar

[17] 17. Titchmarsh, E.C., Introduction to the theory of Fourier integrals (Oxford University Press, Oxford, 1948). Google Scholar

[18] 18. Watson, G.N., A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1958). Google Scholar

[19] 19. Zemanian, A.H., Distribution theory and transform analysis (McGraw Hill Book Co., New York, 1965). Google Scholar

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

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