9.4.E: Problems on Uniform Convergence of Functions and C-Integrals

Exercise $\PageIndex{1}$

Fill in all proof details in Theorems $1-5,$ Corollaries 4 and $5,$ and examples $(\mathrm{A})$ and $(\mathrm{B}) .$

Exercise $\PageIndex{1'}$

Using $(6),$ prove that

$$\lim _{x \rightarrow q} H(x, y) \text { (uniformly) }$$
exists on $B \subseteq E^{1}$ iff
$$(\forall \varepsilon>0)\left(\exists G_{\neg q}\right)(\forall y \in B)\left(\forall x, x^{\prime} \in G_{\neg q}\right) \quad\left|H(x, y)-H\left(x^{\prime}, y\right)\right|<\varepsilon .$$
Assume $E$ complete and $|H|<\infty$ on $G_{\neg q} \times B .$
[Hint: "Imitate" the proof of Theorem 1, using Theorem 2 of Chapter 4, §2.]

Exercise $\PageIndex{2}$

State formulas analogous to ( 1) and ( 2) for $\int_{-\infty}^{a}, \int_{a}^{b-},$ and $\int_{a+}^{b}$.

Exercise $\PageIndex{3}$

State and prove Theorems 1 to 3 and Corollaries 1 to 3 for

$$\int_{-\infty}^{a}, \int_{a}^{b-}, \text { and } \int_{a+}^{b} .$$
In Theorems 2 and 3 explore absolute convergence for
$$\int_{a}^{b-} \text { and } \int_{a+}^{b} .$$
Do at least some of the cases involved.
[Hint: Use Theorem 1 of §3 and Problem 1', if already solved.]

Exercise $\PageIndex{4}$

Prove that

$$\lim _{x \rightarrow q} H(x, y)=F(y) \text { (uniformly) }$$
on $B$ iff
$$\lim _{n \rightarrow \infty} H\left(x_{n}, \cdot\right)=F(\text { uniformly })$$
on $B$ for all sequences $x_{n} \rightarrow q\left(x_{n} \neq q\right)$.
[Hint: "Imitate" Theorem 1 in Chapter 4, §2. Use Definition 1 of Chapter 4, §12.]

Exercise $\PageIndex{5}$

Prove that if

$$\lim _{x \rightarrow q} H(x, y)=F(y) \text { (uniformly) }$$
on $A$ and on $B,$ then this convergence holds on $A \cup B .$ Hence deduce similar propositions on $C$-integrals.

Exercise $\PageIndex{6}$

Show that the integrals listed below violate Corollary 4 and hence do not converge uniformly on $P=(0, \delta)$ though proper L-integrals exist for each $u \in P .$ Thus show that Theorem 1 (ii) does not apply to uniform convergence.
(a) $\int_{0+}^{1} \frac{u d t}{t^{2}-u^{2}}$;
(b) $\int_{0+}^{1} \frac{u^{2}-t^{2}}{\left(t^{2}+u^{2}\right)^{2}} d t$;
(c) $\int_{0+}^{1} \frac{t u\left(t^{2}-u^{2}\right)}{\left(t^{2}+u^{2}\right)^{2}} d t$.
$[\text { Hint for }(\mathrm{b}): \text { To disprove uniform convergence, fix any } \varepsilon, v>0 .$ Then

$$\int_{0}^{v} \frac{u^{2}-t^{2}}{\left(t^{2}+u^{2}\right)^{2}} d t=\frac{v}{v^{2}+u^{2}} \rightarrow \frac{1}{v}$$
as $u \rightarrow 0 .$ Thus if $v<\frac{1}{2 \varepsilon}$,
$$\left.(\exists u \in P) \quad \int_{0}^{v} \frac{u^{2}-t^{2}}{\left(t^{2}+u^{2}\right)^{2}} d t>\frac{1}{2 v}>\varepsilon .\right]$$

Exercise $\PageIndex{7}$

Using Corollaries 3 to $5,$ show that the following integrals converge (uniformly) on $U$ (as listed) but only pointwise on $P$ (for the latter, proceed as in Problem 6 ). Specify $P$ and $M(t)$ in each case where they are not given.
(a) $\int_{0}^{\infty} e^{-u t^{2}} d t ; U=[\delta, \infty) ; P=(0, \delta)$.
$\left.\left[\text { Hint: Set } M(t)=e^{-\delta t} \text { for } t \geq 1 \text { (Corollaries } 3 \text { and } 5\right) .\right]$
(b) $\int_{0}^{\infty} e^{-u t} t^{a} \cos t d t(a \geq 0) ; U=[\delta, \infty)$.
(c) $\int_{0+}^{1} t^{u-1} d t ; U=[\delta, \infty)$.
(d) $\int_{0+}^{1} t^{-u} \sin t d t ; U=[0, \delta], 0<\delta<2 ; P=[\delta, 2) ; M(t)=t^{1-\delta}$.
[Hint: Fix $v$ so small that

$$(\forall t \in(0, v)) \quad \frac{\sin t}{t}>\frac{1}{2} .$$
Then, if $u \rightarrow 2$,
$$\left.\int_{0}^{v} t^{-u} \sin t d t \geq \frac{1}{2} \int_{0}^{v} \frac{d t}{t^{u-1}} \rightarrow \infty .\right]$$

Exercise $\PageIndex{8}$

In example (A), disprove uniform convergence on $P=(0, \infty)$.
[Hint: Proceed as in Problem $6 .]$

Exercise $\PageIndex{9}$

Do example (B) using Theorem 3 and Corollary 5. Disprove uniform convergence on $B .$

Exercise $\PageIndex{10}$

Show that

$$\int_{0+}^{\infty} \frac{\sin t u}{t} \cos t d t$$
converges uniformly on any closed interval $U,$ with $\pm 1 \notin U .$
[Hint: Transform into
$$\left.\frac{1}{2} \int_{0+}^{\infty} \frac{1}{t}\{\sin [(u+1) t]+\sin [(u-1) t]\} d t .\right]$$

Exercise $\PageIndex{11}$

Show that

$$\int_{0}^{\infty} t \sin t^{3} \sin t u d t$$
converges (uniformly) on any finite interval $U$.
[Hint: Integrate
$$\int_{x}^{y} t \sin t^{3} \sin t u d t$$
by parts twice. Then let $y \rightarrow \infty \text { and } x \rightarrow 0 .]$

Exercise $\PageIndex{12}$

Show that

$$\int_{0+}^{\infty} e^{-t u} \frac{\cos t}{t^{a}} d t \quad(0<a<1)$$
converges (uniformly) for $u \geq 0 .$
[Hints: For $\left.t \rightarrow 0+, \text { use } M(t)=t^{-a} . \text { For } t \rightarrow \infty, \text { use example (B) and Theorem 2. }\right]$

Exercise $\PageIndex{13}$

Prove that

$$\int_{0+}^{\infty} \frac{\cos t u}{t^{a}} d t \quad(0<a<1)$$
converges (uniformly) for $u \geq \delta>0,$ but (pointwise) for $u>0 .$
[Hint: Use Theorem 3 with $g(t, u)=\cos t u$ and
$$\left|\int_{0}^{x} g\right|=\left|\frac{\sin x u}{u}\right| \leq \frac{1}{\delta} .$$
For $u>0$,
$$\int_{v}^{\infty} \frac{\cos t u}{t^{a}} d t=u^{a-1} \int_{v u}^{\infty} \frac{\cos z}{z} d z \rightarrow \infty$$
if $v=1 / u \text { and } u \rightarrow 0 . \text { Use Corollary } 4 .]$

Exercise $\PageIndex{14}$

Given $A, B \subseteq E^{1}(m A<\infty)$ and $f: E^{2} \rightarrow E,$ suppose that
(i) each $f(x, \cdot)=f_{x}(x \in A)$ is relatively (or uniformly) continuous on $B ;$ and
(ii) $\operatorname{each} f(\cdot, y)=f^{y}(y \in B)$ is $m$-integrable on $A$.
Set

$$F(y)=\int_{A} f(x, y) d m(x), \quad y \in B.$$
Then show that $F$ is relatively (or uniformly) continuous on $B .$
[Hint: We have
$$\begin{aligned}(\forall x \in A)(\forall \varepsilon>0)\left(\forall y_{0} \in B\right)(&\exists \delta>0)\left(\forall y \in B \cap G_{y_{0}}(\delta)\right) \\ &\left|F(y)-F\left(y_{0}\right)\right| \leq \int_{A}\left|f(x, y)-f\left(x, y_{0}\right)\right| d m(x) \leq \int_{A}\left(\frac{\varepsilon}{m A}\right) d m=\varepsilon . \end{aligned}$$
Similarly for uniform continuity.]

Exercise $\PageIndex{15}$

Suppose that
(a) $C \int_{a}^{\infty} f(t, y) d m(t)=F(y)(\text { uniformly })$ on $B=[b, d] \subseteq E^{1}$;
(b) each $f(x, \cdot)=f_{x}(x \geq a)$ is relatively continuous on $B ;$ and
(c) each $f(\cdot, y)=f^{y}(y \in B)$ is $m$ -integrable on every $[a, x] \subset E^{1},$ $x \geq a .$
Then show that $F$ is relatively continuous, hence integrable, on $B$ and that

$$\int_{B} F=\lim _{x \rightarrow \infty} \int_{B} H_{x} ,$$
where
$$H(x, y)=\int_{a}^{x} f(t, y) d m(t) .$$
(Passage to the limit under the $\int$-sign.)
[Hint: Use Problem 14 and Theorem 4; note that
$$\left.C \int_{0}^{\infty} f(t, y) d m(t)=\lim _{x \rightarrow \infty} H(x, y)(\text {uniformly}) .\right]$$