The Hankel transformation on 𝑀’_{𝜇} and its representation
Authors: Koh E. L. , Li C. K. 
Published: July 2010

The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval’s equation as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis 1 right-parenthesis right-parenthesis EndLabel mathematical left-angle h Subscript mu Baseline f comma phi mathematical right-angle equals mathematical left-angle f comma h Subscript mu Baseline phi mathematical right-angle EndLayout">

(

(
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 }

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis 2 right-parenthesis right-parenthesis EndLabel left-parenthesis h Subscript mu Baseline f right-parenthesis left-parenthesis y right-parenthesis equals mathematical left-angle f left-parenthesis x right-parenthesis comma StartRoot x y EndRoot upper J Subscript mu Baseline left-parenthesis x y right-parenthesis mathematical right-angle period EndLayout">

(

(
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis y right-parenthesis equals left-parenthesis h Subscript mu Baseline f right-parenthesis left-parenthesis y right-parenthesis equals mathematical left-angle f left-parenthesis x right-parenthesis comma StartRoot x y EndRoot upper J Subscript mu Baseline left-parenthesis x y right-parenthesis mathematical right-angle comma f element-of upper M prime Subscript mu">

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 }

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals limit Underscript r right-arrow normal infinity Endscripts integral Subscript 0 Superscript r Baseline upper F left-parenthesis y right-parenthesis StartRoot x y EndRoot upper J Subscript mu Baseline left-parenthesis x y right-parenthesis d y period">

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).