Measure, Integration & Real Analysis, 2021a

Section 11B Fourier Series and L p of Unit Circle 361
EXERCISES 11B
1 Show that the family {e k } k∈Z of trigonometric functions defined by 11.1 is an
orthonormal basis of L 2( (−π, π] ) .
2 Use the result of Exercise 12(a) in Section 11A to show that
1 + 1 3 2 + 1 5 2 + 1 π2
+ ···=
72 8 .
3 Use techniques similar to Example 11.32 to evaluate
∞
∑
n=1
1
n 4 .
[If you feel industrious, you may also want to evaluate ∑ ∞ n=1 1/n6 . Similar
techniques work to evaluate ∑ ∞ n=1 1/nk for each positive even integer k. You can
become famous if you figure out how to evaluate ∑ ∞ n=1 1/n3 , which currently is
an open question.]
4 Suppose f , g : ∂D → C are measurable functions. Prove that the function
(w, z) ↦→ f (w)g(zw) is a measurable function from ∂D × ∂D to C.
[Here the σ-algebra on ∂D × ∂D is the usual product σ-algebra as defined in
5.2.]
5 Where does the proof of 11.42 fail when p = ∞?
6 Suppose f ∈ L 1 (∂D). Prove that f is real valued (almost everywhere) if and
only if ̂f (−n) = ̂f (n) for every n ∈ Z.
7 Suppose f ∈ L 1 (∂D). Show that f ∈ L 2 (∂D) if and only if
∞
∑
n=−∞
| ̂f (n)| 2 < ∞.
8 Suppose f ∈ L 2 (∂D). Prove that | f (z)| = 1 for almost every z ∈ ∂D if and
only if
{
∞
∑
̂f (k) ̂f 1 if n = 0,
(k − n) =
k=−∞
0 if n ̸= 0
for all n ∈ Z.
9 For this exercise, for each r ∈ [0, 1) think of P r as an operator on L 2 (∂D).
(a) Show that P r is a self-adjoint compact operator for each r ∈ [0, 1).
(b) For each r ∈ [0, 1), find all eigenvalues and eigenvectors of P r .
(c) Prove or disprove: lim r↑1 ‖I −P r ‖ = 0.
10 Suppose f ∈ L 1 (∂D). Define T : L 2 (∂D) → L 2 (∂D) by Tg = f ∗ g.
(a) Show that T is a compact operator on L 2 (∂D).
(b) Prove that T is injective if and only if ̂f (n) ̸= 0 for every n ∈ Z.
(c) Find a formula for T ∗ .
(d) Prove: T is self-adjoint if and only if all Fourier coefficients of f are real.
(e) Show that T is a normal operator.
Measure, Integration & Real Analysis, by Sheldon Axler