Real inversion formula for a Hankel type integral transformation (Q1886884)

The authors propose to extend the Hankel type integral transformation \[ J(y)=y^{1+2v} \int^\infty_0 (y\tau)^{-v}J_\mu (y \tau)j (\tau)\,d\tau\quad (0<y< \infty) \] to certain classes of generalized functions where \(J_\mu(z)\) is the Bessel function of first kind of order \(\mu\). A real inversion formula is shown to be valid when the limiting operation in the formula is understood as weak convergence in the space \(D'(I)\) of Schwartz distributions where \(I\) denotes the interval \(0<\tau< \infty\).