Exercise 6, § 3

Let $f:\mathbf{R}\rightarrow \mathbf{R}$ and $g:\mathbf{R}\rightarrow \mathbf{C}$ be two functions. Suppose that $f\in$ $L^{\infty}(\mathbf{R})$ and that $g\in L^1(\mathbf{R}),\int_{\mathbf{R}}g(x)dx= 1$ and that $\mathscr{F}(g)$ does not vanish. Suppose that $f$ is slowly decreasing, that is to say that for $y > x$ such that $y-x$ tends to 0 and $x\rightarrow +\infty$, one has

lim inf($f(y)-f(x)$)$\geqslant 0$.

If $(g*f)(x)$ tends to $\ell$ when $x$ tends to $+\infty$, then $f(x)$ tends to $\ell$ when $x$ tends to $+\infty$.