ANALYSIS QUALIFYING EXAM PROBLEMS BRIAN LEARY ...

ANALYSIS QUALS 41
Problem 8: Let f : R → R be a continuous function that is periodic with
period 2π. Show that if all the Fourier coefficients of f are 0, then f is
identically 0.
Proof. By Parseval, we know that
π
−π
|f(t)| 2 dt = 2π
∞
n=−∞
ˆ
f(n)
By assumption, all Fourier coefficients are 0, so the right-hand side is 0.
Thus,
π
−π
|f(t)| 2 dt = 0.
But this implies that f = 0 a.e., so since f is continuous, we have that f is
identically 0.
Problem 10: Let U be a bounded connected open set in C. Prove that if K is
a compact subset of U, then there is a constant CK such that for every point
z ∈ K and every holomorphic L2 function f on U, |f(z)| ≤ CK
.
2
.
U |f|2 1/2
Solution. Let r = dist(K, U c ). Let z ∈ K, B = B(z, r/2) ⊂ U. By mean
value principle and Cauchy-Schwarz, we have that
|f(z)| =
1
f(w)dw
Area(B)
B
≤ 4
πr2
|f(w)| dw
B
= 4
πr2
|f(w)| χBdw
U
≤ 4
πr2
|f(w)|
U
2 1/2
dw χ
U
2 1/2 Bdw
4
=
πr2 1/2 1/2 |f|
U
Thus, we let CK = 4
πr2 1/2, and note this depends only on K. This
completes the proof.
Problem 11: Let U be a bounded connected open set in C. Suppose that 0
belongs to U and that F : U → U is a holomorphic function with F (0) =
0 and with F ′ (0) = 1. Prove that F is the identity map. [Suggestion:
Consider the power series of F composed with itself many times].
Solution. In a neighborhood of 0, we can write F (z) = z +a2z 2 +a3z 3 +. . ..
Let F (n) (z) = F ◦ F ◦ · · · ◦ F (z), where there are n compositions. Then
F (n) : U → U. We have that
F (2) (z) = F (z) +
akF (z) k = z + 2a2z 2 + 2a3z 3 + . . . .
Similarly, we have that
k≥2
F (n) (z) = z + na2z 2 + na3z 3 + . . . .