The Hankel transformation of Banach-space-valued generalized functions

Abstract:

The object of this paper is to study Banach-space-valued generalized functions belonging to $[{H_\mu }(A);B]$ for which the Hankel transformation may be defined. In Realizability theory for continuous linear systems (Academic Press, New York, 1972), Zemanian considered certain $\rho$-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 ${D^m}(A)$ in the testing function spaces of interest. This method is not possible here since the space ${D^m}(A)$ is not dense in ${H_\mu }(A)$, and the structure of ${H_\mu }(A)$ is quite different from that of ${D^m}(A)$, which has an inductive-limit topology. Thus, it is necessary to introduce a dense subspace ${}_\mu {D_I}(A)$ of ${H_\mu }(A)$ to derive some properties of ${H_\mu }(A)$. We then define the Hankel transformation on $[{H_\mu }(A);B]$. 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.