Exercise 7, § 3

Let $g:\mathbf{R}\rightarrow \mathbf{C}$ be a continuous function such that:

$\sum_{n=-\infty}^{+\infty}$ sup$_{n\leqslant t\leqslant n+1}|g(t)|<+\infty$.

a) Show that $g\in L^1(\mathbf{R})$.

b) Suppose that $g$ has integral 1 and that $\mathscr{F}(g)$ does not vanish. Let $\mu$ be a complex measure on $\mathbf{R}$ such that $|\mu|([x, x+ 1])$ is bounded when $x$ ranges over $\mathbf{R}$. If $(g*\mu)(x)$ tends to $\ell$ when $x$ tends to $+\infty$, then, for every continuous function $h:\mathbf{R}\rightarrow \mathbf{C}$ such that

$\sum_{n=-\infty}^{+\infty}$ sup$_{n\leqslant t\leqslant n+1}|h(t)|<+\infty$,

of integral 1, the limit of $(h*\mu)(x)$ when $x$ tends to $+\infty$ is equal to $\ell$. (Prove that $h*\mu$ is slowly oscillating and that $(g*(h*\mu))(x)$ tends to $\ell$ when $x$ tends to $+\infty .$)