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 F¹(f)(y)= (h_1,α,βf) (f) = ∫₀^∞ x^1+2(α-β) C_α,β (xy) f(x)dx, (α-β ≥ -1/2) i.e. h_1,α,β= F₁ F₂(f)(y) = (h_2,α,βf)(y) = y^1+2(α-β) ∫₀^∞ C_α,β (xy) f (x)dx, (α-β≥-1/2) i.e. h_2,α,β = F₂ 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.