The Hankel transformation on and its representation
Authors:
E. L. Koh and C. K. Li
Journal:
Proc. Amer. Math. Soc. 122 (1994), 10851094
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 , This convergence gives a stronger result than the one obtained by Koh and Zemanian (1968). 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).
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 E. L. Koh and A. H. Zemanian, The complex Hankel and Itransformations of generalized functions, SIAM J. Appl. Math. 16 (1968), 945957.
 [2]
 E. L. Koh, A representation of Hankel transformable generalized functions, SIAM J. Math. Anal. 1 (1970), 3336.
 [3]
 R. E. Edwards, Functional analysis, Holt, Rinehart, and Winston, New York, 1965. MR 0221256 (36:4308)
 [4]
 A. H. Zemanian, Generalized integral transformations, Interscience, New York, 1968. MR 0423007 (54:10991)
 [5]
 E. L. Koh and C. K. Li, The complex Hankel transformation on , Congr. Numer. 87 (1992), 145151. MR 1167657 (93d:44009)
 [6]
 F. Treves, Topological vector spaces distributions and kernels, Academic Press, New York, 1967. MR 0225131 (37:726)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412075397
PII:
S 00029939(1994)12075397
Keywords:
Hankel transformation,
generalized functions,
countable union spaces,
inversion theorem
Article copyright:
© Copyright 1994 American Mathematical Society
