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Let $\mathcal{H}$ be a set of all hyperplanes in $\mathbb{R}^n$. Radon transform of function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ is defined as function $R[f] \colon \mathcal{H} \to \mathbb{R}$ such that $$ R[f](H) = \int\limits_{H} f(x) dS $$ There is a theorem about the support set of function $f$.

Let $f(x) \in C(\mathbb{R}^n)$ satisfy the following conditions:

  1. for any integer $k>0$ function $|x|^k f(x)$ is bounded,
  2. there exists a constant $A>0$ such that if $\mbox{dist }(0,H) > A$ then $R[f](H) = 0$.

Then $f(x) = 0$ if $|x| > A$.

I'm looking for different proofs of this result. One such proof I found in Helgason's book.

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