MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment # 8
Due date: Friday March 26, 2010 (hand-in in class)
This HW assignment has two pages.
Problem 1 (Fejér’s theorem): Do [Rudin, Ch 8. Exercise #15].
Problem 2 (Sobolev imbedding theorem) : Suppose f : R → R is a 2π-periodic Riemann
integrable function.
(a) Assume
∞
X
|n|2 |fˆ(n)|2 < ∞.
−∞
Show that there exists a 2π-periodic continuous function g : R → R such that
Z π
|f (x) − g(x)|2 dx = 0.
−π
(b) Assume for k ∈ N,
∞
X
|n|2(k+1) |fˆ(n)|2 < ∞.
−∞
Show that there exists a 2π-periodic function g : R → R, which is differentiable k times, such
that
Z π
|f (x) − g(x)|2 dx = 0.
−π
Problem 3: Use the same notation and assumptions as in Problem 4 in HW # 7, except now we
assume ONLY that g is Riemann integrable: in particular, g may NOT be differentiable.
Show that the following holds:
Z π
|f (x, t) − g(x)|2 dx → 0 as t → 0+.
−π
Problem 4: (Poincaré inequality)
(a) Show that there exists a constant C > 0 with the following property: Let fR : R → R be a
π
2π-periodic function such that its derivative f 0 exists and integrable. Assume −π f (x)dx = 0.
Then
Z π
Z π
|f 0 (x)|2 dx.
|f (x)|2 dx ≤ C
−π
−π
Find the smallest constant C for which this inequality holds (independently of f ). (Hint: Use
Parseval’s theorem.)
(b) Show that there exists a constant C > 0 with the following property: Let f : R → R be a
differentiable function such that its derivative f 0 is integrable. Assume f (a) = f (b) = 0 for some
a < b ∈ R. Then,
Z
Z
b
b
|f 0 (x)|2 dx.
|f (x)|2 dx ≤ C
a
a
Find the smallest constant C for which this inequality holds (independently of f ). (Hint: Use part
a).)
Essay Problem (Optional, for extra mark (≤ 20)): Hand-in in a separate paper (maximum three
pages). To earn marks your answer should be INTERESTING.
Give an intuitive or physical explanation of the following facts that for 2π-periodic Riemann integrable functions f, g : R → R,
f[
∗ g(n) = fˆ(n)ĝ(n) for all n ∈ Z,
and
fcg(n) =
X
fˆ(n − k)ĝ(k)
for all n ∈ Z.
k∈Z
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. 8, Exercises # 16, # 17, #19.
Problem: Do Rudin, Ch. 8, Exercises # 1. This important exercise is not about Fourier series but
about power series. Use the definition of ex :
∞
X
xk
x
e =
.
k!
k=0
Here 0! = 1.
2