With the notation of Problem 116, show that the only solution f ∈ S(R) of the differential equation a0f(x) + a1∂xf(x) + a2∂ 2 xf(x) + …+ a2∂

The equality occurs in (36) if and only if f is a translation or a modulation of a normalized

Gaussian e−ax 2 , a > 0.

Problem 104. Define a dilation of a function f : R→ C by

ft(x) = t − 1

2f(t−1x) (t > 0) .

Show that

‖ ft ‖2=‖ f ‖2 (t > 0) .

36 T. PRZEBINDA

We compute

‖ ft ‖22= ∫ R |ft(x)|2 dx =

∫ R t−1|f(t−1x)|2 dx =

∫ R |f(x)|2 dx .

Problem 105. Show that

F(ft) = (Ff)t−1 (t > 0) .

We compute

F(ft)(y) = ∫ R e−2πixyt−

1 2f(t−1x) dx =

∫ R e−2πixtyt

1 2f(x) dx = (Ff)t−1(y) .

Problem 106. Show that

σ(ft)σ̃(ft) = σ(f)σ̃(f) (t > 0) .

We compute

µ(ft) =

∫ R x|t−

1 2f(t−1x)|2 dx =

∫ R tx|f(x)|2 dx = tµ(f)

σ2(ft) =

∫ R (x− µ(ft))2|ft(x)|2 dx =

∫ R (tx− tµ(f))2|f(x)|2 dx = t2σ2(f) .

Similarly, we see from Problem 105 that

σ̃2(ft) = t −2σ̃2(f) .

Hence the formula follows.

5. Exam 2, due Monday 11/26/2018 in class.

Problem 107. Let k = 0, 1, 2, …. Suppose f ∈ L1(R/Z) is k – times differentiable (i.e. the derivatives f ′, f ′′, …, f (k) exist and belong to f ∈ L1(R/Z). Show that there is a constant C such that

|Ff(n)| ≤ C(1 + |n|)k (n ∈ Z) .

Problem 108. Let k = 0, 1, 2, …. Suppose f ∈ L1(R) is k – times differentiable (i.e. the derivatives f ′, f ′′, …, f (k) exist and are integrable, i.e. belong to f ∈ L1(R). Show that there is a constant C such that

|Ff(y)| ≤ C(1 + |y|)k (y ∈ R) .

MATH 4123, HOMEWORK, EXAMS AND SOLUTIONS, FALL 2018 37

Problem 109. Show that ∞∑ n=1

1

n4 = π4

90 .

Hint: use Parseval’s formula and Problem 61.

Problem 110. Compute the integral∫ R

( 1

π

1

x2 + 1

)2 dx .

Hint: use Parseval’s formula and Lemma 2.4 on page 150 in the text.

Problem 111. Give an example of a non-zero function f ∈ L1(R/Z) is such that f ∗f = f .

Problem 112. Suppose f ∈ L1(R) is such that f ∗ f = f . Show that f = 0.

Problem 113. Let p(x) be a polynomial and let

f(x) =

{ p(x−1)e−x

−1 if x > 0 ,

0 if x ≤ 0 .

Show that the derivative f ′(0) = lim x→0

f(x)−f(0) x

exists and is equal to 0.

Problem 114. Let

φ(x) =

{ e−x

−1 if x > 0 ,

0 if x ≤ 0 . Show that the derivative φ(k)(0) exists and is equal to 0, and therefore φ(k)(x) exists for any x ∈ R.

Problem 115. Show that the space C∞c (R) is non-trivial in the sense that it contains a non-zero function.

38 T. PRZEBINDA

Problem 116. Let f, g ∈ S(R) and let a0, a1, …, aN ∈ C be constants. Show that the functions f and g satisfy the differential equation

a0f(x) + a1∂xf(x) + a2∂ 2 xf(x) + …+ a2∂

N x f(x) = g(x) (x ∈ R)

if and only if their Fourier transforms f̂ and ĝ satisfy the the following algebraic equation,

(a0 + a1(2πiy) + a2(2πiy) 2 + …+ a2(2πiy)

N)f̂(y) = ĝ(y) (y ∈ R) .

Problem 117. With the notation of Problem 116, show that the only solution f ∈ S(R) of the differential equation

a0f(x) + a1∂xf(x) + a2∂ 2 xf(x) + …+ a2∂

N x f(x) = 0 (x ∈ R) ,

where not all an are zero, is f = 0.