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

DOI:
https://doi.org/10.1090/S0002-9939-1993-1149972-7

MathSciNet review:
1149972

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Abstract: The object of this paper is to study Banach-space-valued generalized functions belonging to for which the Hankel transformation may be defined. In *Realizability theory for continuous linear systems* (Academic Press, New York, 1972), Zemanian considered certain -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 in the testing function spaces of interest. This method is not possible here since the space is not dense in , and the structure of is quite different from that of , which has an inductive-limit topology. Thus, it is necessary to introduce a dense subspace of to derive some properties of . We then define the Hankel transformation on . 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.

**[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****[2]**A. H. Zemanian,*Realizability theory for continuous linear systems*, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 97. MR**0449807****[3]**A. K. Tiwari,*Banach space valued distributional Mellin transform and form invariant linear filtering*, Indian J. Pure Appl. Math.**20**(1989), no. 5, 493–504. MR**1000064****[4]**E. L. Koh,*The Hankel transformation of negative order for distributions of rapid growth*, SIAM J. Math. Anal.**1**(1970), 322–327. MR**0267395**, https://doi.org/10.1137/0501028**[5]**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,*Higher transcendental functions*, Vol. II, McGraw-Hill, New York, 1953.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1149972-7

Keywords:
The Hankel transformation,
inductive-limit topology,
generalized functions,
Banach space

Article copyright:
© Copyright 1993
American Mathematical Society