Exercise 1, § 3

Let I be an arbitrary interval in $\mathbf{R}$, let A be a set, and g a finite real function defined on $I \times A$ such that for every $\alpha \in A$ the map $t \mapsto g(t, \alpha)$ is decreasing and $\geq 0$ on I; suppose further that there is a number M independent of $\alpha$ such that $g(t, \alpha) \leq M$ on $I \times A$.

a) Show that if $f$ is a regulated function on I such that the integral $\int_I f(t) \, dt$ converges, then the integral $\int_I f(t) g(t, \alpha) \, dt$ is uniformly convergent for $\alpha \in A$ (use the second mean value theorem; cf. II, p. 82, exerc. 16).

b) Suppose that $I = [a, +\infty[$ and also that $g(t, \alpha)$ tends uniformly to 0 (for $\alpha \in A$) when $t$ tends to $+\infty$. Show that if $f$ is a regulated function on I and if there is a number $k > 0$ such that $\| \int_J f(t) \, dt \| \leq k$ for every compact interval J contained in I, then the integral $\int_I f(t) g(t, \alpha) \, dt$ is uniformly convergent for $\alpha \in A$ (same method).

c) Suppose that $I = [a, +\infty[$ with $a > 0$, $A = [0, +\infty[$, and $g(t, \alpha) = \varphi(\alpha t)$, where $\varphi$ is a convex decreasing function and $\geq 0$ on A, tending to 0 as $x$ tends to $+\infty$ and such that $\varphi(0) = 1$. Suppose further that $f = h''$, where $h$ is a twice differentiable function on $[a, +\infty[$, and, together with $h'$, is bounded on this interval. The integral $\int_a^{+\infty} f(t) \varphi(\alpha t) \, dt$ is then convergent for $\alpha > 0$: show that when $\alpha$ tends to 0, it tends to $-h'(a)$ (integrate by parts and use the second mean value theorem).

d) Take $\varphi(x) = e^{-cx}$, where $c > 0$, take $a = 1$, and for $f$ take the complex function $t \mapsto e^{t \log t}/t$, which is the derivative of a bounded function on I; show that for a suitable choice of $c$ the integral $\int_1^{+\infty} e^{u \alpha t} \frac{e^{t \log t}}{t} \, dt$, which is absolutely convergent for $\alpha > 0$, does not tend to any limit when $\alpha$ tends to 0 (after integrating by parts, make the change of variables $\alpha t = u$; use Laplace transform theory to see that $\int_0^{+\infty} e^{-cu + t \log u} \, du$ does not vanish for certain $c > 0$).