9.4.E: Problems on Uniform Convergence of Functions and C-Integrals
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fill in all proof details in Theorems 1−5, Corollaries 4 and 5, and examples (A) and (B).
Using (6), prove that
limx→qH(x,y) (uniformly)
exists on B⊆E1 iff
(∀ε>0)(∃G¬q)(∀y∈B)(∀x,x′∈G¬q)|H(x,y)−H(x′,y)|<ε.
Assume E complete and |H|<∞ on G¬q×B.
[Hint: "Imitate" the proof of Theorem 1, using Theorem 2 of Chapter 4, §2.]
State formulas analogous to ( 1) and ( 2) for ∫a−∞,∫b−a, and ∫ba+.
State and prove Theorems 1 to 3 and Corollaries 1 to 3 for
∫a−∞,∫b−a, and ∫ba+.
In Theorems 2 and 3 explore absolute convergence for
∫b−a and ∫ba+.
Do at least some of the cases involved.
[Hint: Use Theorem 1 of §3 and Problem 1', if already solved.]
Prove that
limx→qH(x,y)=F(y) (uniformly)
on B iff
limn→∞H(xn,⋅)=F( uniformly )
on B for all sequences xn→q(xn≠q).
[Hint: "Imitate" Theorem 1 in Chapter 4, §2. Use Definition 1 of Chapter 4, §12.]
Prove that if
limx→qH(x,y)=F(y) (uniformly)
on A and on B, then this convergence holds on A∪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,δ) though proper L-integrals exist for each u∈P. Thus show that Theorem 1 (ii) does not apply to uniform convergence.
(a) ∫10+udtt2−u2;
(b) ∫10+u2−t2(t2+u2)2dt;
(c) ∫10+tu(t2−u2)(t2+u2)2dt.
[ Hint for (b): To disprove uniform convergence, fix any ε,v>0. Then
∫v0u2−t2(t2+u2)2dt=vv2+u2→1v
as u→0. Thus if v<12ε,
(∃u∈P)∫v0u2−t2(t2+u2)2dt>12v>ε.]
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) ∫∞0e−ut2dt;U=[δ,∞);P=(0,δ).
[ Hint: Set M(t)=e−δt for t≥1 (Corollaries 3 and 5).]
(b) ∫∞0e−uttacostdt(a≥0);U=[δ,∞).
(c) ∫10+tu−1dt;U=[δ,∞).
(d) ∫10+t−usintdt;U=[0,δ],0<δ<2;P=[δ,2);M(t)=t1−δ.
[Hint: Fix v so small that
(∀t∈(0,v))sintt>12.
Then, if u→2,
∫v0t−usintdt≥12∫v0dttu−1→∞.]
In example (A), disprove uniform convergence on P=(0,∞).
[Hint: Proceed as in Problem 6.]
Do example (B) using Theorem 3 and Corollary 5. Disprove uniform convergence on B.
Show that
∫∞0+sintutcostdt
converges uniformly on any closed interval U, with ±1∉U.
[Hint: Transform into
12∫∞0+1t{sin[(u+1)t]+sin[(u−1)t]}dt.]
Show that
∫∞0tsint3sintudt
converges (uniformly) on any finite interval U.
[Hint: Integrate
∫yxtsint3sintudt
by parts twice. Then let y→∞ and x→0.]
Show that
∫∞0+e−tucosttadt(0<a<1)
converges (uniformly) for u≥0.
[Hints: For t→0+, use M(t)=t−a. For t→∞, use example (B) and Theorem 2. ]
Prove that
∫∞0+costutadt(0<a<1)
converges (uniformly) for u≥δ>0, but (pointwise) for u>0.
[Hint: Use Theorem 3 with g(t,u)=costu and
|∫x0g|=|sinxuu|≤1δ.
For u>0,
∫∞vcostutadt=ua−1∫∞vucoszzdz→∞
if v=1/u and u→0. Use Corollary 4.]
Given A,B⊆E1(mA<∞) and f:E2→E, suppose that
(i) each f(x,⋅)=fx(x∈A) is relatively (or uniformly) continuous on B; and
(ii) eachf(⋅,y)=fy(y∈B) is m-integrable on A.
Set
F(y)=∫Af(x,y)dm(x),y∈B.
Then show that F is relatively (or uniformly) continuous on B.
[Hint: We have
(∀x∈A)(∀ε>0)(∀y0∈B)(∃δ>0)(∀y∈B∩Gy0(δ))|F(y)−F(y0)|≤∫A|f(x,y)−f(x,y0)|dm(x)≤∫A(εmA)dm=ε.
Similarly for uniform continuity.]
Suppose that
(a) C∫∞af(t,y)dm(t)=F(y)( uniformly ) on B=[b,d]⊆E1;
(b) each f(x,⋅)=fx(x≥a) is relatively continuous on B; and
(c) each f(⋅,y)=fy(y∈B) is m -integrable on every [a,x]⊂E1, x≥a.
Then show that F is relatively continuous, hence integrable, on B and that
∫BF=limx→∞∫BHx,
where
H(x,y)=∫xaf(t,y)dm(t).
(Passage to the limit under the ∫-sign.)
[Hint: Use Problem 14 and Theorem 4; note that
C∫∞0f(t,y)dm(t)=limx→∞H(x,y)(uniformly).]