Exercise 7, § 1

Let $f\in L^1(G)$. Let F be a holomorphic function defined in an open subset U of $\mathbf{C}$. Suppose that U contains the set of values of $\mathscr{F}(f)$, and if G is not discrete, suppose that $0\in U$ and F(0) = 0.

Then there exists $g\in L^1(G)$ such that $F\circ \mathscr{F}(f) =\mathscr{F}(g)$. (Use the holomorphic functional calculus.)