Measure, Integration & Real Analysis, 2021a

Section 11C Fourier Transform 379
15 Define f ∈ L 1 (R) by f (x) =e −x4 χ [0, ∞)
(x). Show that ̂f /∈ L 1 (R).
16 Suppose f ∈ L 1 (R) and ̂f ∈ L 1 (R). Prove that f ∈ L 2 (R) and ̂f ∈ L 2 (R).
17 Prove there exists a continuous function g : R → R such that lim g(t) =0
t→±∞
and g /∈ {̂f : f ∈ L 1 (R)}.
18 Prove that if f ∈ L 1 (R), then ‖ ̂f ‖ 2 = ‖ f ‖ 2 .
[This exercise slightly improves Plancherel’s Theorem (11.82) because here we
have the weaker hypothesis that f ∈ L 1 (R) instead of f ∈ L 1 (R) ∩ L 2 (R).
Because of Plancherel’s Theorem, here you need only prove that if f ∈ L 1 (R)
and ‖ f ‖ 2 = ∞, then ‖ ̂f ‖ 2 = ∞.]
19 Suppose y > 0. Define on operator T on L 2 (R) by Tf = f ∗ P y .
(a) Show that T is a self-adjoint operator on L 2 (R).
(b) Show that sp(T) =[0, 1].
[Because the spectrum of each compact operator is a countable set (by 10.93),
part (b) above implies that T is not a compact operator. This conclusion differs
from the situation on the unit circle—see Exercise 9 in Section 11B.]
20 Prove that if f ∈ L 1 (R) and g ∈ L 2 (R), then F( f ∗ g) = ̂f ·Fg.
21 Prove that f , g ∈ L 2 (R), then ( fg)̂ =(F f ) ∗ (F g).
Measure, Integration & Real Analysis, by Sheldon Axler