The generalized Hankel-Clifford transformation on M′α and its representation

The generalized hankel-clifford transformations defined by F
₁
(y) = (h
_1,α,β
f) (y) = y
^-(α+β)
∫
₀
^∞
ℐ
_j,β
(xy)f(x) dx, with h
_1,α,β
= h
_1,α,β·
^-1
= ∫
₀
^∞
(y/x)
^-(α+β)/2
J
_α-β
(2(√xy))f(x)dx and F
₂
(y) = (h
_2α,β
f) (y) = ∫
₀
^∞
x
^-(α+β)
ℐ
_α,β
(xy)f (x)dx, with h
_2α,β
= h
_2,α,β
^-1
= ∫
₀
^∞
(x/y)
^-(α+β)/2
Jα-β(2√(xy))f(x)dx where ℐ
_α,β
(z) =z
^(α,β)/2
J
_α-β
(2√(z)), J
_α-β
(z) being the Bessel function of the first kind and order (α - β) ≥ -1/ 2 were extended by Malgonde [8] to certain generalized functions and by Malgonde and Bandewar [7] to certain generalized functions of slow growth through a generalization of mixed Parseval's equation as 〈h'
_1,α,β
f,φ〉 = 〈f, h'
_2,α,β
φ〉, where f ∈ H'
_α
(I) and φ ∈ H
_α
(I). Note that by setting φ=h
_2,α,β
Φ where Φ, h
_2,α,β
Φ ∈ H
_α
(I). Analogously, we can define the distributional transformation h'
_2,α,β
on H'
_β
(I) as the adjoint of h'
_1,α,β
on H'
_β
(I) as. Thus equation takes the form 〈h'
_1,α,β
f, h'
_2,α,β
φ〉 = 〈f, Φ〉, where f ∈ H'
_α
, Φ ∈ H
_α
and in this form equation appears as a generalization of the mixed Parseval equation ∫
₀
^∞
f(x)g(x)dx = ∫
₀
^∞
F
₁
(y)G
₂
(y)dy, where F
₁
(y) = (h
_1,α,β
f)(y) and g(x) = (h
_2,α,β
^-1
G
₂
(y))(x), as it happens in the extension of the Hankel-Clifford transform to a certain spaces of generalized functions studied in Malgonde [8] and Mendez [5]. Letter, Malgonde [8] defined the distributional generalized Hankel-Clifford transformation of the conventional transformation (I) for complex y on H
_α,β
= H
_α,β
(σ) = ∪
_n=1
^∞
H
_α,β,α
n
, where H
_α,β,α
n
is the testing function space whick contains the kernel function, y
^-α-β
(xy)
^(α+β)/2
J
_α-β
(2√(xy)). The distributional complex generalized -Hankel-Clifford transformation of the conventional generalized-Hankel-Clifford transformation (I) on H
_α,β
= H
_α,β
(σ) = ∪
_n=1
^∞
H
_α,β,α
n
, where H
_α,β
is the testing space which contains the kernel function, y
^-α-β
(xy)
^(α+β)/2
J
_α-β
(2√(xy)), is possible for equatuib (I) and (II) similar to Koh and Zemanian [2] and Malgonde and Lakshmi Gorty [9]. The distributional transformation was defined directly as the application of a generalized function to the kernel function, i.e., for f ∈ H'
_α,β
, F'
₁
(y) = (h'
_1,α,β
f)(y) = 〈f(x),y
^-α-β
(xy)
^(α+β)/2
J
_α-β
(2√(xy))〉. In this paper, we extend definition (IV) to a larger space of generalized functions. We first introduce the test function space M
_{α, a}
which contains the kernel function and show that H
_α
⊂ M
_α,a
⊂ H
_α,β,a
. We then form the countable union space M
_α
= ∪
_n=1
^∞
M
_α,a
n
whose dual M'
_α
has H'
_α,β
as a subsapce. Our main result is an inversion theorem states as follows. Let F'
₁
(y) = (h'
_1,α,β
f)(y) = 〈f(x), y
^-α-β
(xy)
^(α+β)/2
J
_α-β
(2√(xy))〉, f ∈ M'
_α
where y is restrcited to the positive real axis. Let (α-β) ≥ -½. Then, in the sense of convergence in H'
_α
, f(x) = lim
_r→∞
∫
₀
^∞
F'
₁
(y) y
^-α-β
ℐ
_α,β
(xy)dy. This convergence gives a stronger result than the one obtained by Malgonde [8]. Secondly, we prove that every generalized function belonging to M'
_α,a
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 [6] and similar to one by Koh [3].