Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A mixed Parseval equation and the generalized Hankel transformations

Author: J. M. Méndez
Journal: Proc. Amer. Math. Soc. 102 (1988), 619-624
MSC: Primary 46F12; Secondary 44A05, 47B35
MathSciNet review: 928991
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {T_1}$ and $ {T_2}$ be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation

$\displaystyle \int_0^\infty {f(x)g(x)dx = \int_0^\infty {{F_1}(y){G_2}(y)} \;dy,} $

where $ {F_1}(y) = ({T_1}f)(y)$ and $ {G_2}(y) = ({T_2}g)(y)$. We propose to define the generalized transformation $ {T'_1}$ as the adjoint operator of $ {T_2}$, and conversely. This procedure provides a new approach to extend the Hankel transform to certain spaces of distributions.

References [Enhancements On Off] (What's this?)

  • [1] E. L. Koh and A. H. Zemanian, The complex Hankel and $ I$-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968), 945-957.
  • [2] W. Y. Lee, On Schwartz's Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal. 6 (1975), 427-432. MR 0361766 (50:14211)
  • [3] D. P. Misra and J. L. Lavoine, Transforms analysis of generalized functions, North-Holland, Amsterdam, 1986. MR 832188 (88i:46052)
  • [4] I. N. Sneddon, The use of integral transforms, Tata McGraw-Hill, New Delhi, 1974.
  • [5] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford Univ. Press, London, 1959.
  • [6] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1958. MR 1349110 (96i:33010)
  • [7] A. H. Zemanian, A distributional Hankel transformation, J. SIAM Appl. Math. 14 (1966), 561-576. MR 0201930 (34:1807)
  • [8] -, A solution of a division problem arising from Bessel type differential equations, J. SIAM Appl. Math. 15 (1967), 1106-1111. MR 0221228 (36:4280)
  • [9] -, Generalized integral transformations, Interscience, New York, 1968.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46F12, 44A05, 47B35

Retrieve articles in all journals with MSC: 46F12, 44A05, 47B35

Additional Information

Keywords: Hankel transformation, Parseval equation, generalized functions, Bessel operator
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society