  Let $F$ be a local field and $n\ge 2$ an integer. We study the Radon
transform as an operator $M : \mathcal C_+ \to \mathcal C_-$ from the space of
smooth $K$-finite functions on $F^n \setminus \{0\}$ with bounded support to
the space of smooth $K$-finite functions on $F^n \setminus \{0\}$ supported
away from a neighborhood of $0$. These spaces naturally arise in the theory of
automorphic forms. We prove that $M$ is an isomorphism and provide formulas for
$M^{-1}$. In the real case, we show that when $K$-finiteness is dropped from
the definitions, the analog of $M$ is not surjective.
