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

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 μ f , φ ⟩ = ⟨ f , h μ φ ⟩ \begin{equation}\tag {$(1)$} \langle {h_\mu }f,\varphi \rangle = \langle f,{h_\mu }\varphi \rangle \end{equation} where φ , h μ φ ∈ H μ , f ∈ H μ ′ \varphi , {h_\mu }\varphi \in {H_\mu }, f \in {H’_\mu } . Later, Koh and Zemanian defined the generalized complex Hankel transformation on J μ = ⋃ ν = 1 ∞ J a ν , μ {J_\mu } = {\bigcup }_{\nu = 1}^\infty \,{J_{{a_\nu },\mu }} , where J a ν , μ {J_{{a_\nu },\mu }} is the testing function space which contains the kernel function, x y J μ ( x y ) \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 ∈ J μ ′ f \in {J’_\mu } , ( ( 2 ) ) ( h μ f ) ( y ) = ⟨ f ( x ) , x y J μ ( x y ) ⟩ . \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 , μ {M_{a,\mu }} which contains the kernel function and show that H μ ⊂ M a , μ ⊂ J a , μ {H_\mu } \subset {M_{a,\mu }} \subset {J_{a,\mu }} . We then form the countable union space M μ = ⋃ ν = 1 ∞ M a ν , μ {M_\mu } = {\bigcup }_{\nu = 1}^\infty \,{M_{{a_\nu },\mu }} whose dual M μ ′ {M’_\mu } has J μ ′ {J’_\mu } as a subspace. Our main result is an inversion theorem stated as follows. Let F ( y ) = ( h μ f ) ( y ) = ⟨ f ( x ) , x y J μ ( x y ) ⟩ , f ∈ M μ ′ 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 μ ≥ − 1 2 \mu \geq - \frac {1}{2} . Then, in the sense of convergence in H μ ′ {H’_\mu } , \[ f ( x ) = lim r → ∞ ∫ 0 r F ( y ) x y J μ ( x y ) d y . f(x) = \lim \limits _{r \to \infty } \int _0^r {F(y)} \sqrt {xy} {J_\mu }(xy)dy. \] 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 , μ ′ {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).