The Hankel transformation on and its representation

Authors:
E. L. Koh and C. K. Li

Journal:
Proc. Amer. Math. Soc. **122** (1994), 1085-1094

MSC:
Primary 46F12; Secondary 44A15

MathSciNet review:
1207539

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Abstract: The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval's equation as

() |

where .

Later, Koh and Zemanian defined the generalized complex Hankel transformation on , where is the testing function space which contains the kernel function, . A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for ,

() |

In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space which contains the kernel function and show that . We then form the countable union space whose dual has as a subspace. Our main result is an inversion theorem stated as follows.

Let , where *y* is restricted to the positive real axis. Let . Then, in the sense of convergence in ,

Secondly, we prove that every generalized function belonging to can be represented by a finite sum of derivatives of measurable functions. This proof is analogous to the method employed in structure theorems for Schwartz distributions (Edwards, 1965), and similar to one by Koh (1970).

**[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]**E. L. Koh,*A representation of Hankel transformable generalized functions*, SIAM J. Math. Anal.**1**(1970), 33-36.**[3]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR**0221256****[4]**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****[5]**E. L. Koh and C. K. Li,*The complex Hankel transformation on 𝑀’_{𝜇}*, Proceedings of the Twenty-first Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1991), 1992, pp. 145–151. MR**1167657****[6]**François Trèves,*Topological vector spaces, distributions and kernels*, Academic Press, New York-London, 1967. MR**0225131**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1207539-7

Keywords:
Hankel transformation,
generalized functions,
countable union spaces,
inversion theorem

Article copyright:
© Copyright 1994
American Mathematical Society