I'm trying to prove the following theorem: The following are equivalent for a locally compact abelian group:
$G$ has an open compact subgroup.
There exists a nonzero $f\in C_c(G)$ such that $\hat{f}\in C_c(\hat{G})$.
To prove that (1) implies (2), I let $H$ be a compact open subgroup of $G$ and let $f=\chi_H$ be the characteristic function of $H$. Then of course $\hat{f}=\chi_{H^\perp}$ where $H^\perp$ is the set of all characters in $\hat{G}$ which are identically $1$ on $H$. Compactness of $H^\perp\cong \widehat{G/H}$ is not so clear to me. The openness assumption should be used here I suppose.
The other direction seems to be far more complicated. If somebody can outline a proof or give a reference to a source where I can read the proof I would appreciate it!