Exercise 19, § 3

For $f\in L^1(\mathbf{R})$, let $A_f$ denote the closed subalgebra of $L^1(\mathbf{R})$ generated by $f$. (a) There exists $f\in L^1(\mathbf{R})$ such that $A_f$ is a maximal proper subalgebra of $L^1(\mathbf{R})$.

Suppose in the sequel that $f\in L^1(\mathbf{R})$ is a function such that $A_f$ is a maximal proper subalgebra of $L^1(\mathbf{R})$.

(b) Let $S =\{0\} \cup \widehat{f}(\mathbf{R})\subset \mathbf{C}$. Now prove that the function $z\mapsto \overline{z}$ is not the uniform limit on S of polynomial functions. (First prove that $\mathbf{C}-$ S is not connected.)

(c) The subalgebra of $L^1(\mathbf{R})$ generated by $f$ and $\widetilde{f}$ is dense in $L^1(\mathbf{R})$. (d) The Fourier transform of $f$ is injective and does not vanish on $\mathbf{R}$. (Prove that S cannot have two bounded connected components.)