Exercise 3, § 3

Let $g$ be a complex function defined on $]0,+\infty [$, integrable for Lebesgue measure. We suppose that $\int_0^{+\infty}g(t)dt= 1$ and that $\int_0^{+\infty}g(t)t^{ix}dt\not= 0$ for all $x\in \mathbf{R}$. Let $f$ be a measurable and bounded complex function in $]0,+\infty [$, such that $\frac{1}{x}\int_0^{+\infty}g(t/x)f(t)dt$ tends to a finite limit $\ell$ when $x$ tends to $+\infty$.

a) For every complex function $h$ defined on $]0,+\infty [$, integrable for Lebesgue measure and of integral 1, one has

lim$_{x\rightarrow+\infty}\frac{1}{x}\int_0^{+\infty}h(\frac{t}{x})f(t)dt=\ell$.

In particular,

lim$_{x\rightarrow+\infty}\frac{1}{x}\int_0^xf(t)dt=\ell$.

(Setting $g_1(t) =tg(t),h_1(t) =th(t)$, we have $g_1\in L^1(\mathbf{R}_+^*)$; the Fourier transform $\mathscr{F}(g_1)$ does not vanish and $\breve{g}_1*f$ tends to $\ell$, hence $h_1*f$ tends to $\ell .$) b) If $f(t)$ is slowly oscillating on $\mathbf{R}^*_+$ when $t$ tends to $+\infty$, then $f$ tends to $\ell$ when $t$ tends to $+\infty$.