Distributional generalized Hankel-Schwartz type transformations on the spaces L′p,v of distributions

The published paper by Waphare [12] is not correct as the transforms defined by the author is incorrect and the results obtained are totally wrong. So in this paper we study the behavior of the generalized Hankel-Schwartz type transformations depending on real parameters α and β defined by F
¹
(f)(y)= (h
_1,α,β
f) (f) = ∫
₀
^∞
x
^1+2(α-β)
C
_α,β
(xy) f(x)dx, (α-β ≥ -1/2) i.e. h
_1,α,β
= F
₁
F
₂
(f)(y) = (h
_2,α,β
f)(y) = y
^1+2(α-β)
∫
₀
^∞
C
_α,β
(xy) f (x)dx, (α-β≥-1/2) i.e. h
_2,α,β
= F
₂
where C
_αβ
(z) = z
^-(α-β)
J
_α-β
(z), J
_α-β
being the Bessel function of the first kind and order α - β, on the spaces L
_p,v
introduced by P.G.Rooney [8] and the Hankel transformation was studied by P.G.Rooney in [9,10]. However the approach followed by Rooney is essentially different to the method used by us in this paper. We also extend these transformations to L'
_p,v
the space of distributions.