MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment # 7
Due date: Friday March 19, 2010 (hand-in in class)
Problem 1: Suppose f : R → R is 2π-periodic, i.e. f (x + 2π) = f (x) for all x ∈ R. Assume f
is differentiable infinitely many times, i.e. all k-th derivatives f (k) exists, k = 1, 2, 3, · · · . Assume
further that there exists M > 0 such that |f (k) | ≤ M for all k ∈ N. Show that f is of the form
f (x) = A cos x + B sin x + C
where A, B, C ∈ R are constants.
Problem 2: Do [Rudin, Ch8. Exercise # 13].
Problem 3: We have the following Riemann-Lebesgue Lemma:
Riemann-Lebesgue Lemma: Let g : R → R be a 2π-periodic function, i.e. g(x + 2π) = g(x)
for all x ∈ R. Assume g ∈ R (i.e. Riemann integrable in the sense of [Rudin]). Then
|ĝ(n)| → 0 as n → ∞.
(a) Derive this lemma from Parseval’s theorem.
(b) Prove this WITHOUT using Parseval’s theorem or Stone-Weierstrass theorem. (Use only the
properties of Riemann integrals.)
Problem 4: (Heat equation with periodic initial data) Let g : R → R be a 2π-periodic function,
i.e. g(x + 2π) = g(x) for all x ∈ R. Assume g is differentiable and g 0 is Riemann integrable (in
the sense of [Rudin]).
(a) Find a formula for the solution f : R × [0, ∞) → R to the following initial value problem of
heat equation
(
∂2
∂
f (x, t) = ∂x
2 f (x, t) ,
∂t
f (x, 0) = g(x) for x ∈ R .
(You may assume that the solution f is 2π-periodic in x-variable.)
(b) Show that the above solution f satisfies that f (x, t) → g(x) as t → 0+, uniformly in x.
(c) (smoothing of the heat equation) Show that for fixed t > 0, the above solution f (x, t) is
differentiable in x infinitely many times.
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 # 12, # 14.
*The following two questions are difficult and beyond the scope of the course: you can ignore
them.
Problem*:( Hölder continuity and convergence of Fourier series) A function f : R → R is
said to be Hölder continuous with Hölder exponent δ for 0 < δ ≤ 1, if there exists M > 0 such
that for any x 6= y ∈ R,
|f (x) − f (y)|
≤M
|x − y|δ
Show that for a 2π-periodic function f : R → R, if f is Hölder continuous with exponent δ,
0 < δ ≤ 1, then the Fourier series
∞
X
fˆ(n)einx
−∞
converges uniformly to f .
Problem*:(Hölder continuity and convergence of Fourier series) Show that if a 2π-periodic
function f : R → R is Hölder continuous with exponent δ, 1/2 < δ ≤ 1, then
∞
X
|fˆ(n)| < ∞.
−∞
In particular,
∞
X
fˆ(n)einx
−∞
converges uniformly to f . (Note that this convergence is stronger than the previous Problem*.)
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