9.4.E: Problems on Uniform Convergence of Functions and C-Integrals
Fill in all proof details in Theorems \(1-5,\) Corollaries 4 and \(5,\) and examples \((\mathrm{A})\) and \((\mathrm{B}) .\)
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.]
State formulas analogous to ( 1) and ( 2) for \(\int_{-\infty}^{a}, \int_{a}^{b-},\) and \(\int_{a+}^{b}\).
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.]
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.]
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.
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 \(v<\frac{1}{2 \varepsilon}\),
\[
\left.(\exists u \in P) \quad \int_{0}^{v} \frac{u^{2}-t^{2}}{\left(t^{2}+u^{2}\right)^{2}} d t>\frac{1}{2 v}>\varepsilon .\right]
\]
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 \(u \rightarrow 2\),
\[
\left.\int_{0}^{v} t^{-u} \sin t d t \geq \frac{1}{2} \int_{0}^{v} \frac{d t}{t^{u-1}} \rightarrow \infty .\right]
\]
In example (A), disprove uniform convergence on \(P=(0, \infty)\).
[Hint: Proceed as in Problem \(6 .]\)
Do example (B) using Theorem 3 and Corollary 5. Disprove uniform convergence on \(B .\)
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
\[
\left.\frac{1}{2} \int_{0+}^{\infty} \frac{1}{t}\{\sin [(u+1) t]+\sin [(u-1) t]\} d t .\right]
\]
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 \(y \rightarrow \infty \text { and } x \rightarrow 0 .]\)
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 \(\left.t \rightarrow 0+, \text { use } M(t)=t^{-a} . \text { For } t \rightarrow \infty, \text { use example (B) and Theorem 2. }\right]\)
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 \(v=1 / u \text { and } u \rightarrow 0 . \text { Use Corollary } 4 .]\)
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.]
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
\[
\left.C \int_{0}^{\infty} f(t, y) d m(t)=\lim _{x \rightarrow \infty} H(x, y)(\text {uniformly}) .\right]
\]