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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Hankel transformation of Banach-space-valued generalized functions


Authors: E. L. Koh and C. K. Li
Journal: Proc. Amer. Math. Soc. 119 (1993), 153-163
MSC: Primary 46F12; Secondary 44A15
MathSciNet review: 1149972
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Abstract: The object of this paper is to study Banach-space-valued generalized functions belonging to $ [{H_\mu }(A);B]$ for which the Hankel transformation may be defined. In Realizability theory for continuous linear systems (Academic Press, New York, 1972), Zemanian considered certain $ \rho $-type testing function spaces for which the Laplace transformation is defined. Tiwari (Banach space valued distributional Mellin transform and form invariant linear filtering, Indian J. Pure Appl. Math. 20 (1989), 493-504) follows Zemanian in extending the Mellin transform. Their works are based on the denseness of the Schwartz space $ {D^m}(A)$ in the testing function spaces of interest. This method is not possible here since the space $ {D^m}(A)$ is not dense in $ {H_\mu }(A)$, and the structure of $ {H_\mu }(A)$ is quite different from that of $ {D^m}(A)$, which has an inductive-limit topology. Thus, it is necessary to introduce a dense subspace $ {}_\mu {D_I}(A)$ of $ {H_\mu }(A)$ to derive some properties of $ {H_\mu }(A)$. We then define the Hankel transformation on $ [{H_\mu }(A);B]$. We end this paper with some operational formulas, which are analogous with those given by the first author in SIAM J. Math. Anal. 1 (1970), 322-327.


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

  • [1] A. H. Zemanian, Generalized integral transformations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1968. Pure and Applied Mathematics, Vol. XVIII. MR 0423007 (54 #10991)
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1149972-7
PII: S 0002-9939(1993)1149972-7
Keywords: The Hankel transformation, inductive-limit topology, generalized functions, Banach space
Article copyright: © Copyright 1993 American Mathematical Society