The Hankel transformation on 𝑀’_{𝜇} and its representation

Abstract

Abstract. The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval's equation as (1) (h"f, 9) = (/, K<t)where tp , h^y eH¡¡, f e H^.Later, Koh and Zemanian defined the generalized complex Hankel transformation on J/, = (JI^Li Jov.it> where Jov.it is the testing function space which contains the kernel function, yf^yJit(xy) ■ A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for feJp,(hitf)(y) = (f(x),Vxyjfl(xy)).In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space Ma,ii which contains the kernel function and show that Hf, c Ma,p¡ C Ja,it ■ We then form the countable union space M^ = \J%X May,n whose dual M^ has J^ as a subspace. Our main result is an inversion theorem stated as follows.