National Polytechnic School

Second year of preparatory class

2024-2025
Mathematical analysis 4
Tutorial serie n ◦ 07– Fourier Transforms

Exercise 1.
Consider the function f given by

1,
 0 < x < 1,
f (x) = 1/2, x = 1,

0, x > 0.


1. Determine the Fourier cosine and sine transforms. 2. Deduce the Fourier integral representations in cosine and sine.
What can be concluded ?

Exercise 2.
Let f be the function defined by

−x
e , x > 0,

f (x) = 1/2, x = 0,

0, x < 0.


1. Calculate the Fourier coefficients of f . 2. Verify, using Dirichlet’s convergence theorem, that we have

−x
 ∞ πe , x > 0,

cos(ωx) + ω sin(ωx)
dω = α, x = 0,
0 1 + ω2 
β, x < 0,


where α and β are to be determined.

Exercise 3.
Establish, using Fourier integral representations, each of the following relations :
1.  ∞ 3
ω sin(ωx) π
4+4
dω = e−x cos x, x > 0.
0 ω 2
2.  (
∞ π
cos(πω/2) cos(ωx) 2 cos x, |x| < π/2,
dω =
0 1 − ω2 0, |x| ≥ π/2.
3.  (
∞ π
sin(πω) sin(ωx) 2 sin x, 0 < x ≤ π,
dω =
0 1 − ω2 0, x > π.

Exercise 4.
Consider the function fα (x) = e−α|x| , where α is a strictly positive real parameter and x ∈ R.
1. Determine the associated Fourier transform. 2. Deduce the value of the Laplace integral of the first kind given by
 ∞
cos(ωx)
dω.
0 α2 + ω 2

3. Define the function gα by


−fα (x),
 x < 0,
gα (x) = fα (x), x > 0,

0, x = 0.


1
 (a) Determine the Fourier transform of gα .
(b) Deduce the value of the Laplace integral of the second kind given by
 ∞
ω sin(ωx)
dω.
0 α2 + ω 2

4. (a) By direct calculation, find the values of the integrals

 ∞
dω
0 (α2 + ω 2 )2

and  ∞
ω2
dω.
0 (α2 + ω 2 )2
(b) Retrieve the same results using Parseval’s equality.

Exercise 5.
Let f be the function given by
(
x2 , |x| < 1,
f (x) =
0, |x| ≥ 1.
1. Determine the Fourier transform F (ω) = F (f (x)). 2. Using the inverse transform, show that we have

 ∞ x2 , |x| < 1,
 2  cos(ωx) π
(ω − 2) sin ω + 2ω cos ω dω = 1/2, |x| = 1,
0 ω3 2
0, |x| > 1.


3. Using Parseval’s equality, calculate the integral

 ∞
1  2 2
(ω − 2) sin ω + 2ω cos ω dω.
0 ω6

Exercise 6.
Let f be the function defined by
(
1 − |x|, |x| ≤ 1,
f (x) =
0, |x| > 1.
1. Verify that f satisfies the conditions of Dirichlet’s theorem and then plot its graph. What can be observed ?
2. (a) Using Dirichlet’s convergence theorem, show that for all ξ ∈ R
 ∞
sin2 λ
cos(λξ)dλ = αf (βξ),
0 λ2

where the coefficients α and β are to be determined.

(b) Deduce the values of the integrals
 ∞
sin2 λ
dλ,
0 λ2
 ∞
sin2 λ cos λ
dλ,
0 λ2
 ∞
sin2 λ cos(2λ)
dλ.
0 λ2
3. (a) Determine the Fourier transform of the function f .
(b) Using Parseval’s equality, deduce the value of the integral
 ∞
sin4 ω
dω.
0 ω4

2
Exercise 7.
Consider the function f defined by

1,
 |x| < 1,
f (x) = 1/2, |x| = 1,

0, |x| > 1.


1. Determine the Fourier transform of this function. 2. Calculate, using two different methods, the transform of the
function h = f ∗ f . 3. Deduce, using Parseval’s equality, the values of the following two integrals :
 ∞
sin2 ω
dω,
0 ω2
 ∞
sin4 ω
dω.
0 ω4
Recall that the convolution product of f by g is defined by
 ∞
(f ∗ g)(x) = f (t)g(x − t)dt.
−∞

Exercise 8.
Consider the three functions f, g and h defined by

 
1 2 3 2
 8 (3 − x ), |x| ≤ 1, − 16 (1 + x ), |x| ≤ 1,
(
1
+ x2 ), |x| < 3,
 
1 2
f (x) = 16 (x − 6x + 9), 1 < |x| < 3, 3
g(x) = − 8 |x|, 16 (9
1 < |x| < 3, h(x) =
  0, |x| ≥ 3.
0, |x| ≥ 3. 0, |x| ≥ 3.
 

1. Plot these three functions on the same graph. 2. Verify that for all x ∈ R, f (x) = g(x) + h(x). 3. Determine G and H ;
the Fourier transforms of the functions g and h respectively, i.e., G(ω) = F (g(x)) and H(ω) = F (h(x)), and then deduce
that of f , i.e., F (ω) = F (f (x)). 4. Prove that for all x ∈ R
 ∞
sin3 ω
cos(ωx)dω = πf (x).
0 ω3

∞ 3
∞ sin3 ω cos ω

∞ sin3 ω cos(2ω)
∞ sin3 ω cos(3ω)
5. Deduce the values of the following integrals : (a) 0 sinω3 ω dω, (b) dω, (c) dω, (d) dω.

∞ 6 0 ω3 0 ω3 0 ω3
6. Using Parseval’s formula, calculate the integral 0 sinω6 ω dω.

Exercise 9.
2
Consider the function fα (x) = e−αx , where x ∈ R and α is a strictly positive real parameter. Let Fα be the associated
Fourier transform, i.e., Fα (ω) = F (fα (x)).
1. Prove that the function Fα satisfies the ordinary differential equation
ω
Fα0 (ω) + Fα (ω) = 0. (1)
2α
Hint : Interchange the sign of differentiation with respect to ω and that of generalized integration with respect to x
without justification.
2
2. Deduce that Fα (ω) = e−c /4α , determining the constant c. 3. (a) Using the convergence theorem, show that for all
x∈R  ∞
2 √
e−c /4α cos(ωx)dω = απe−αx2 .
0

(b) Verify that applying Parseval’s equality to this function leads to a trivial result.

3
Exercise 10.
Consider the ordinary differential equation with constant coefficients given by

−y 00 + y = g(x), (2)

where the unknown function y is such that lim|x|→∞ y (k) (x) = 0, k = 0, 1.

G(ω)
1. Prove that we have Y (ω) = 1+ω 2 , where Y (ω) = F (y(x)) and G(ω) = F (g(x)). 2. Using the Fourier transform of the

convolution product, show that the solution y is obtained by the integral formula
 ∞
y(x) = α g(t)e−|x−t| dt,
−∞

determining the coefficient α.  

Hint : Use the Laplace integral of the first kind (cf. Exercise 4) to calculate F −1 1+ω
1
2 .
(
1, |x| < 1,
3. Deduce the solution of equation (2) for the case g(x) =
0, |x| ≥ 1.