Measure, Integration & Real Analysis, 2021a

378 Chapter 11 Fourier Analysis
3 Suppose f (x) =4πx 2 e −πx2 − e −πx2 for all x ∈ R. Show that ̂f = − f .
4 Find f ∈ L 1 (R) such that f ̸= 0 and ̂f = if.
5 Prove that if p is a polynomial on R with complex coefficients and f : R → C
is defined by f (x) =p(x)e −πx2 , then there exists a polynomial q on R with
complex coefficients such that deg q = deg p and ̂f (t) =q(t)e −πt2 for all
t ∈ R.
6 Suppose
f (x) =
{
xe −2πx if x > 0,
0 if x ≤ 0.
Show that ̂f 1
(t) =
4π 2 for all t ∈ R.
(1 + it) 2
7 Prove the formulas in 11.55 for the Fourier transforms of translations, rotations,
and dilations.
8 Suppose f ∈ L 1 (R) and n ∈ Z + . Define g : R → C by g(x) =x n f (x). Prove
that if g ∈ L 1 (R), then ̂f is n times continuously differentiable on R and
for all t ∈ R.
( ̂f ) (n) (t) =(−2πi) n ĝ(t)
9 Suppose n ∈ Z + and f ∈ L 1 (R) is n times continuously differentiable and
f (n) ∈ L 1 (R). Prove that if t ∈ R, then
( f (n) )̂(t) =(2πit) n ̂f (t).
10 Suppose 1 ≤ p ≤ ∞, f ∈ L p (R), and g ∈ L p′ (R). Prove that f ∗ g is a
uniformly continuous function on R.
11 Suppose f ∈L ∞ (R), x ∈ R, and f is continuous at x. Prove that
lim(P y f )(x) = f (x).
y↓0
12 Suppose p ∈ [1, ∞] and f ∈ L p (R). Prove that P y (P y ′ f )=P y+y ′ f for all
y, y ′ > 0.
13 Suppose p ∈ [1, ∞] and f ∈ L p (R). Prove that if 0 < y < y ′ , then
14 Suppose f ∈ L 1 (R).
‖P y f ‖ p ≥‖P y ′ f ‖ p .
(a) Prove that ( f )̂(t) = ̂f (−t) for all t ∈ R.
(b) Prove that f (x) ∈ R for almost every x ∈ R if and only if ̂f (t) = ̂f (−t)
for all t ∈ R.
Measure, Integration & Real Analysis, by Sheldon Axler