# Hankel Transform - Eigenfunctions and Inverse

I was reading Akhiezer's Lectures on Integral Transforms and in chapter nine, The Hankel Transform, he says that because the kernel of the Hankel transform is symmetric, its eigenfunctions corresponding to different eigenvalues must be orthogonal. Can anyone furnish a proof of this or point me in the right direction so that I can prove it myself? I've read a few other texts on integral transforms, but none mentioned this fact or anything near it.

He also mentions that for $\nu > -1$, given some restrictions on a test function $f$ (how quickly it decays), that $$f(y) = \int_0^{\infty} dr \sqrt{yr} J_{\nu}(ry) \int_0^{\infty} dx f(x) \sqrt{xr} J_{\nu}(xr) .$$ He leaves the proof to the reader, but I am completely unable to even see where to begin proving that this is the case. After extensive Googling, I feel I've come up short. Does anyone have any idea how this can be proved?

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Welcome to MSE Sometimes. I edited your post (apparently, we have to surround TeX with "\$" here); please make sure I didn't muck it up. –  David Mitra Jan 9 '12 at 17:03
I don't know anything about the Hankel transform, but a symmetric kernel should give you a self-adjoint operator, and the eigenfunctions of any self-adjoint operator on a Hilbert space are orthogonal. –  Akhil Mathew Jan 9 '12 at 17:04
Thanks, David. I'm used to using Tex The World, so I was unaware. –  Cameron Williams Jan 9 '12 at 17:08
Akhil, that is exactly what I was looking for. Thanks a ton! It's been two years since I took PDE, so that fact had slipped my mind. –  Cameron Williams Jan 9 '12 at 17:15