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) .
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.