Analytic representation of the distributional finite Hankel transform (Q579594)

Following \textit{A. H. Zemanian} [Siam J. Appl. Math. 14, 678-690 (1966; Zbl 0154.138)] who defined the Hankel transformation of a distribution of rapid growth in the space \(B'_{\mu}\) which is dual of \(B_{\mu}\), the authors define the testing function spaces \(B_{\mu,b}\) which yield \(B_{\mu}\) as a strict inductive limit as b tends to infinity through a monotonically increasing sequence of positive numbers. The fact that the functions in \(B_{\mu,b}\) are identically zero after b, is used to define the finite Hankel transformation for the generalized functions in its dual \(B'_{\mu,b}\). This is done by generalizing Parseval's equation. It is shown that for \(\mu\geq -1/2\), the finite Hankel transform \(h_{\mu}\) maps \(B'_{\mu,b}\) isomorphically onto the generalized function space \(Y'_{\mu,b}\). The testing function spaces \(B_{\mu,b}\), \(B'_{\mu,b}\), \(Y_{\mu,b}\) and \(Y'_{\mu,b}\) are studied in detail. Numbers of representations of finite Hankel transform of generalized functions are obtained. An inversion theorem for this transform is also established which yields another respresentation of the members of \(B'_{\mu,b}\) as a Fourier-Bessel series. Numbers of examples are given to illustrate the results of the paper.