MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment # 6
Due date: Monday March 1, 2010 (hand-in in class)
Problem 1: (Convolution) Let f, g : R → R be two bounded compactly supported functions.
(Here, f is called compactly supported if there exists a compact set K such that f (x) = 0 for all
x∈
/ K. The smallest closed such set K is called the support of f .) Assume that f , g are Riemann
integrable. Define the convolution f ∗ g : R → R as
Z ∞
f (t)g(x − t)dt
f ∗ g(x) =
−∞
(a) Show that f ∗ g is also compactly supported. Suppose f , g are supported on the intervals [a, b],
[c, d], respectively. What can you say about the support of f ∗ g?
(b) Assume further that either f or g is continuous. Show f ∗ g is continuous.
(c) Let h : R → R be a compactly supported bounded Riemann integrable function. Recall
the sup norm metric d∞ defined on the space of bounded functions on R, that is d∞ (f, g) =
supx∈R |f (x) − g(x)|. Show that
Z ∞
|h|.
d∞ (f ∗ h, g ∗ h) ≤ d∞ (f, g)
−∞
In particular, if fn → f uniformly, then fn ∗ h → f ∗ h uniformly.
Problem 2: (Convolution) In Problem 1 (b), remove the assumption of continuity of f or g. Show
that f ∗g is still continuous. In fact, there is a two line proof of this using the Lebesquge dominated
convergence theorem (see [Rudin 11.32], which is beyond the scope. In this problem, you are NOT
allowed to use Lebesgue dominated convergence theorem. Instead, use the following steps:
(a) First verify that it is enough to show that for each x, y ∈ R,
Z ∞
|g(x − t) − g(y − t)|dt → 0, as y → x.
−∞
In the following we will show this convergence.
Fix an arbitrary > 0.
(b) Show that there exists a function h of the form
h(x) =
k
X
ci χ[ai ,bi ] (x), ai < bi , ci ∈ R
i=1
such that
Z
∞
|g(t) − h(t)|dt ≤ .
−∞
Here, χE denotes the characteristic function of the set E; i.e. χE (x) = 1 for x ∈ E, and χE (x) = 0
for x ∈
/ E.
(c) Show that for each closed interval [a, b]
Z ∞
|χ[a,b] (x − t) − χ[a,b] (y − t)|dt → 0, as y → x.
−∞
(d) Show that we can choose δ > 0 such that for any |x − y| ≤ δ,
Z ∞
|g(x − t) − g(y − t)|dt ≤ .
−∞
Problem 3: Do [Rudin, Ch7. Exercise # 21].
Problem 4: Prove the following:
(a) If f is a continuous real-valued function defined on the set [a, b] × [c, d] and > 0, then there
exists a real-valued polynomial function p in two variables such that |f (x, y) − p(x, y)| < for all
x in [a, b] and y in [c, d].
(b) If X and Y are two compact metric spaces and f : X × Y → R is a continuous function,
then for every > 0 there exist n > 0 and real-valued continuous functions f1 , f2 , ..., fn on X and
real-valued continuous functions g1 , g2 , ..., gn on Y such that
n
X
sup f (x, y) −
fi (x)gi (y) < .
(x,y)∈X×Y
i=1
The following are suggested exercises. Please DO NOT hand-in, but, it is important for you to
do these suggested exercises!
Problem: Do Rudin, Ch. 7, Exercises # 22.
Problem*: (Convolution) This exercise is beyond the scope of this course.
In Problem 1, further assume that f is differentiable.
(a) Use Lebesgue Dominated Convergence Theorem (see [Rudin, 11.32]) to show very easily that
f ∗ g is differentiable and its derivative satisfies
(f ∗ g)0 = f 0 ∗ g.
(Especially, if f is smooth, i.e. differentiable at any order, then f ∗ g is also smooth.)
(b)* Show this WITHOUT using Lebesgue Dominated Convergence Thereom (LDCT).
Problem*: (Density of the set of nowhere differentiable continuous functions) Let N D denote the set of nowhere differentiable continuous functions on the unit interval [0, 1]. That is, for
every function f ∈ N D, f is NOT differentiable at any point in [0, 1]. Is N D a dense subset in
C([0, 1], R)? (Here, C([0, 1], R) denotes the space of real-valued continuous functions on [0, 1]
equipped with the sup norm metric d∞ .) Justify your answer!
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