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The Hankel transformation on $ M'\sb \mu$ 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

$\displaystyle \langle {h_\mu }f,\varphi \rangle = \langle f,{h_\mu }\varphi \rangle$ ($ (1)$)

where $ \varphi , {h_\mu }\varphi \in {H_\mu }, f \in {H'_\mu }$.

Later, Koh and Zemanian defined the generalized complex Hankel transformation on $ {J_\mu } = {\bigcup}_{\nu = 1}^\infty\,{J_{{a_\nu },\mu }} $, where $ {J_{{a_\nu },\mu }}$ is the testing function space which contains the kernel function, $ \sqrt {xy} {J_\mu }(xy)$. A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for $ f \in {J'_\mu }$,

$\displaystyle ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle .$ ($ (2)$)

In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space $ {M_{a,\mu }}$ which contains the kernel function and show that $ {H_\mu } \subset {M_{a,\mu }} \subset {J_{a,\mu }}$. We then form the countable union space $ {M_\mu } = {\bigcup}_{\nu = 1}^\infty\,{M_{{a_\nu },\mu }} $ whose dual $ {M'_\mu }$ has $ {J'_\mu }$ as a subspace. Our main result is an inversion theorem stated as follows.

Let $ F(y) = ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle ,f \in {M'_\mu }$, where y is restricted to the positive real axis. Let $ \mu \geq - \frac{1}{2}$. Then, in the sense of convergence in $ {H'_\mu }$,

$\displaystyle f(x) = \mathop {\lim }\limits_{r \to \infty } \int_0^r {F(y)} \sqrt {xy} {J_\mu }(xy)dy.$

This convergence gives a stronger result than the one obtained by Koh and Zemanian (1968).

Secondly, we prove that every generalized function belonging to $ {M'_{a,\mu }}$ 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).

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] E. L. Koh, A representation of Hankel transformable generalized functions, SIAM J. Math. Anal. 1 (1970), 33-36.
  • [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 $ {M'_\mu }$, Congr. Numer. 87 (1992), 145-151. 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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Keywords: Hankel transformation, generalized functions, countable union spaces, inversion theorem
Article copyright: © Copyright 1994 American Mathematical Society

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