Let
T
1
{T_1}
and
T
2
{T_2}
be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation
\[
∫
0
∞
f
(
x
)
g
(
x
)
d
x
=
∫
0
∞
F
1
(
y
)
G
2
(
y
)
d
y
,
\int _0^\infty {f(x)g(x)dx = \int _0^\infty {{F_1}(y){G_2}(y)} \;dy,}
\]
where
F
1
(
y
)
=
(
T
1
f
)
(
y
)
{F_1}(y) = ({T_1}f)(y)
and
G
2
(
y
)
=
(
T
2
g
)
(
y
)
{G_2}(y) = ({T_2}g)(y)
. We propose to define the generalized transformation
T
1
′
{T’_1}
as the adjoint operator of
T
2
{T_2}
, and conversely. This procedure provides a new approach to extend the Hankel transform to certain spaces of distributions.