The Hankel transformation on $M’_ \mu$ and its representation
HTML articles powered by AMS MathViewer
- by E. L. Koh and C. K. Li PDF
- Proc. Amer. Math. Soc. 122 (1994), 1085-1094 Request permission
Abstract:
The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval’s equation as \begin{equation}\tag {$(1)$} \langle {h_\mu }f,\varphi \rangle = \langle f,{h_\mu }\varphi \rangle \end{equation} 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 $f \in {J’_\mu }$, \begin{equation}\tag {$(2)$} ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle .\end{equation} 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 ${H’_\mu }$, \[ f(x) = \lim \limits _{r \to \infty } \int _0^r {F(y)} \sqrt {xy} {J_\mu }(xy)dy.\] 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
-
E. L. Koh and A. H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968), 945-957.
E. L. Koh, A representation of Hankel transformable generalized functions, SIAM J. Math. Anal. 1 (1970), 33-36.
- R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
- A. H. Zemanian, Generalized integral transformations, Pure and Applied Mathematics, Vol. XVIII, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. MR 0423007
- E. L. Koh and C. K. Li, The complex Hankel transformation on $M’_\mu$, Proceedings of the Twenty-first Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1991), 1992, pp. 145–151. MR 1167657
- François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 1085-1094
- MSC: Primary 46F12; Secondary 44A15
- DOI: https://doi.org/10.1090/S0002-9939-1994-1207539-7
- MathSciNet review: 1207539