Exercise 2, § 3

Let $F$ be a closed subset of $\widehat{G}$ and K a compact subset of $\widehat{G}$ such that $F\cap K =\emptyset$. Prove that there exists a continuous function $g$ integrable on G such that $\mathscr{F}(g)$ is equal to 0 on F and to 1 on K. (Let $f\in L^1(G)$ such that $\mathscr{F}(f)$ is zero on F and equal to 1 on K; consider $f*\varphi$, where $\varphi$ is the Fourier transform of a function of the form $h*h'$ with $h\in \mathscr{K}(\widehat{G}),h'\in \mathscr{K}(\widehat{G})$, and where $h*h'= 1$ on K.)