A key technical step in the proof of the Radon Transform support theorem is proving that the result holds for radial functions. The proof presented in class was elementary but technical and, in my opinion, bewildering. Not the sort of thing I’d come up with.

Yes, I know, I need to work on my skills.

Instead of doing that, I came up with another proof that I find more satisfying. It is fairly simple, and uses nothing more than the Paley-Wiener theorem and a bit of Calculus.

**Theorem** if , , , and is radial, then has compact support and .

The rest of this post will be the proof. Here is a quick sketch:

- is analytic of exponential type in , by the Paley-Wiener Theorem.
- , so the exponential type bounds we got in for in (1) transfer to bounds on the pure partial derivatives of
- If is radial, then at the origin

for all multi-indices - This shows that the mixed partials of are dominated by the pure partials at the origin, and the exponential type estimates in (2) hold for all partials
- So is analytic of exponential type. Thus has compact support. Continue reading