290 CHAPTER 11. FUNDAMENTAL TRANSFORMS
11.7 Exercises1. For f ∈ L1 (Rn), show that if F−1 f ∈ L1 or F f ∈ L1, then f equals a continuous
bounded function a.e. This is Theorem 11.6.11 but review it.
2. Suppose f ,g ∈ L1(R) and F f = Fg. Show f = g a.e.
3. ↑ Suppose f ∗ f = f or f ∗ f = 0 and f ∈ L1(R). Show f = 0.
4. Let h(x) =(∫ x
0 e−t2dt)2
+
(∫ 10
e−x2(1+t2)1+t2 dt
). Show that h′ (x) = 0 and h(0) = π/4.
Then let x→∞ to conclude that∫
∞
0 e−t2dt =
√π/2. Show that
∫∞
−∞e−t2
dt =√
π and
that∫
∞
−∞e−ct2
dt =√
π√c .
5. Let h(x) =(∫ x
0 e−t2dt)2
. Then
h′ (x) = 2(∫ x
0e−t2
dt)
e−x2= 2xe−x2
(∫ 1
0e−(xu)2
du).
Now h(x) =∫ x
0 h′ (t)dt. Do integration by parts to obtain
−e−t2∫ 1
0e−(tu)
2du|x0−
∫ x
0e−t2
∫ 1
0e−(tu)
22tu2dudt
= −e−x2∫ 1
0e−(xu)2
du +1−∫ x
0
∫ 1
0e−t2(1+u2)2tu2dudt
= −e−x2∫ 1
0e−(xu)2
du +1−∫ 1
0u2∫ x
0e−t2(1+u2)2tdtdu
= e(x)+1−∫ 1
0u2
(−e−x2(1+u2)−1
1+u2
)du
= e(x)+1−∫ 1
0
u2
1+u2 du
where limx→∞ e(x) = 0. Now explain why 1−∫ 1
0u2
1+u2 du = 14 π. Hence
∫∞
0 e−t2dt =
√π
2 .
6. Recall that for f a function, fy (x) = f (x−y) . Find a relationship between F fy (t)and F f (t) given that f ∈ L1 (Rn).
7. For f ∈ L1 (Rn) , simplify F f (t+y) .
8. For f ∈ L1 (Rn) and c a nonzero real number, show F f (ct) = Fg(t) where g(x) =f( x
c
).
9. Suppose that f ∈ L1 (R) and that∫|x| | f (x)|dx < ∞. Find a way to use the Fourier
transform of f to compute∫
x f (x)dx.
10. Suppose f ∈ G . Go over why F( fx j)(t) = it jF f (t).