I was reading through Akhiezer's book *Lectures on Integral Transforms* and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$,


$f(y) = \int\_0^{\infty}dr \sqrt{ry}J\_{\nu}(ry) \int\_0^{\infty} dx f(x) \sqrt{xr} J\_{\nu}(xr)$.


He leaves the proof that it is unitary to the reader, but I cannot for the life of me even begin to know what the angle of attack should be. I've done some pretty extensive Googling, but have come up completely short. Can anyone point me in the right direction or if you know where to find the proof itself, can link me to it? Much appreciated.


I have found a couple of proofs for the case where $\nu > -1/2$, but none for the range $-1 < \nu < -1/2$.



Inverse Hankel Transform
<http://en.wikipedia.org/wiki/Hankel_transform#Orthogonality>


This ensures orthogonality. You should think in terms of linear algebra - transition matrix to any orthogonal basis in orthogonal. So you need to prove orhogonality of Bessels. This is stated in Wiki Link above.


PS


Actually I do not quite understand where this orthogonality comes from ?
Can one comment ?


Because usually orthogonality comes easily - the eigenfunctions of some symmetric matrix (operator) are orthogonal.
But Bessels are eigenfucntions in $\nu$ , not in "x". So I am puzzled.


