Final Exam
Math 501
Due December 19, 2014
Instructions: This is a take-home exam. You must complete all of the
problems below and turn in your solutions in Carver 492 by 12pm on the due
date unless prior arrangements are made. You may consult only your textbook,
your lecture notes, the lecture notes on the class web page, and me (Tim).
Problem 1
Let f : [0, 1] × R → R be a bounded, continuous function and let a ∈ R. Let
n ≥ 1. For each k = 0, . . . , n, let tk = k/n. Define a piecewise linear, continuous
function xn : [0, 1] → R by
a
t = t0
xn (t) =
xn (tk−1 ) + f (tk−1 , xn (tk−1 ))(t − tk−1 ) t ∈ (tk−1 , tk ],
for each k = 1, . . . , n.
(a) Prove that there exists a constant M > 0 such that |xn (t)| ≤ M for all n
and all t ∈ [0, 1].
(b) Prove that the sequence (xn ) is equicontinuous.
(c) Prove that there exists a subsequence (xnk ) of (xn ) which converges uniformly to a continuous function x : [0, 1] → R.
(d) Let
∆n (t) = x0n (t) − f (t, xn (t))
for all t ∈ [0, 1] \ {t0 , . . . , tn } and ∆n (t) = 0 for t ∈ {t0 , . . . , tn }. Prove that
∆n (t) converges uniformly to 0 on [0, 1].
(e) Prove that f (t, xnk (t)) converges uniformly to f (t, x(t)) on [0, 1]. [Hint: use
the fact that f is uniformly continuous on [0, 1] × [−M, M ].]
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(f) Prove that x(t) solves the integral equation
Z t
f (s, x(s)) ds
x(t) = a +
0
and hence solves the initial value problem x0 (t) = f (t, x(t)), x(0) = a. [Hint:
Rt
note that xn (t) = a + 0 x0n (s) ds and use the previous two parts.]
Problem 2
Let k : [0, 1] × [0, 1] → R be continuous. For each Riemann integrable
function f : [0, 1] → R, define T f to be the function on [0, 1] given by
Z 1
(T f )(x) =
k(x, y)f (y) dy.
0
(a) Prove that if f is continuous, then T f is continuous.
(b) Prove that if B ⊂ C[0, 1] is a bounded set of functions (that is, there exists a
constant M > 0 such that kf k∞ ≤ M for all f ∈ B), then T (B) is compact
in C[0, 1]. [Hint: use Arzela-Ascoli.]
Problem 3
Let X be a compact metric space , let Lip(X) be the set of R-valued Lipschitz
functions defined on X, and let C(X) be the set of all R-valued continuous
functions defined on X.
(a) Let f, g ∈ Lip(X) and let c ∈ R. Prove that f + cg ∈ Lip(X) and f g ∈
Lip(X).
(b) Prove that for each pair of distinct points x, y ∈ X, there exists f ∈ Lip(X)
such that f (x) 6= f (y). [Hint: see Problem 3 of the midterm.]
(c) Prove that Lip(X) is dense in C(X) with respect to the sup-metric.
Problem 4
Let f : [a, b] → R be a continuous function such that
Z
b
xk f (x) dx = 0
a
for all k ≥ 0.
2
(a) Prove that
b
Z
p(x)f (x) dx = 0
a
for any polynomial p.
(b) Prove that
Z
b
f (x)2 dx = 0.
a
[Hint: use the Weierstrass approximation theorem.]
(c) Prove that f ≡ 0 on [a, b].
Problem 5
Let f : [0, 2π] → R be a function.
(a) Prove that if f is continuously differentiable– that is, differentiable with a
continuous derivative– then
Z 2π
lim
f (x) cos(nx) dx = 0.
(1)
n→±∞
0
[Hint: use integration by parts.]
(b) Prove that if f is continuous, then (1) still holds. [Hint: you guessed it–
use Weierstrass approximation.]
Remark. Part (b) is a special case of the celebrated Riemann-Lebesgue Lemma
of harmonic analysis– not to be confused with our earlier result, the so-called
Riemann-Lebesgue theorem, which characterized the set of Riemann integrable
functions.
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