Some Exercises on Fourier Analysis∗
General Properties of the Fourier Modes
Exercise 1. Let f : T → C be an integrable function. Prove the following:
i. f (x ) ∼ fˆ (0) +
P
h
n ≥1
i
fˆ (n ) + fˆ (−n ) cos(nx ) + i fˆ (n ) − fˆ (−n ) sin(nx ) .
ii. If f is even then fˆ (n ) = fˆ (−n ) and we get a cos series.
iii. If f is odd then fˆ (n ) = fˆ (−n ) and we get a sin series.
iv. If f (x + π ) = f (x ) for all x ∈ R, then fˆ (n ) = 0 for all n odd.
v. f (x ) ∈ R for a.e. x iff fˆ (n ) = fˆ (−n ), for every n ∈ Z.
Exercise 2. Let fk , f : T → C, k ≥ 1 be integrable functions such that
Z
π
|fk (x ) − f (x )| dx = 0.
lim
(2.A)
k →∞
−π
Prove that fˆk (n ) → fˆ (n ) uniformly in n.
Decay of the Fourier Modes
Exercise 3. Let f : T → C be a square integrable function. Show that
fˆ (n ) ∼ o (1).
(3.A)
Exercise 4. Let f ∈ Ck (T), k ∈ N. Show that
fˆ (n ) ∼ O
(4.A)
!
1
.
|n |k
Is it true that fˆ (n ) ∼ o 1/|n |k ?
Exercise 5. Let f : T → R be a bounded monotone function. Show that
!
ˆf (n ) ∼ O 1 .
|n |
(5.A)
Exercise 6. Let f : T → C be an integrable function. Show that
i.
Z
1
fˆ (n ) = −
(6.A)
2π
π
f x+
−π
π
n
e−inx dx,
hence
(6.B)
1
fˆ (n ) =
2π
Z
π
f (x ) − f x +
−π
π
n
e−inx dx.
ii. If f ∈ Cα (T), 0 < α < 1, then
(6.C)
fˆ (n ) ∼ O
1
|n |α
!
.
∗ The document contains material I collected myself while I was running the support classes on Fourier Analysis during
the academic year 2015-2016. Though it might cover several aspects of the module, it should not be considered a
representative of the module’s exam.
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iii. If f (x ) =
P∞
k =0
k
2−kα ei2 x , then f ∈ Cα (T) and fˆ (n ) =
for all n = 2k .
1
,
n
Exercise 7. We have already proved the following:
i. If f ∈ Ck (T), k ≥ 1, then fˆ (n ) ∼ o (1/|n |k ).
ii. If f ∈ Cα (T), 0 < α < 1, then fˆ (n ) ∼ O (1/|n |α ).
iii. If f is bounded and monotone, then fˆ (n ) ∼ O (1/|n |).
iv. If f is integrable, then fˆ (n ) ∼ O (1).
Show that for every sequence {εn }n ∈N that converges to 0, there exists f : T → R such that
fˆ (n ) ≥ εn ,
(7.A)
for infinitely many n.
Convergence of the Fourier Series
Exercise 8. Let f ∈ C2 (T). Show that SN f → f uniformly.
Exercise 9. Let f ∈ C1 (T). Show that SN f → f uniformly.
Exercise 10. Let f : T → C and x ∈ T such that
(10.A)
f (x + ) := lim+ f (y), f (x − ) := lim− f (y) < ∞.
y →x
y→x
Show that
σN f (x ) →
(10.B)
f (x + ) + f (x − )
,
2
where σN f is the Cesaro mean of Sn f , i.e. σN f = (1/N )
PN −1
n =0
Sn f (or σN f = FN ∗ f ).
Integral Computations
Exercise 11. Let FN be the N -th Fejer kernel, i.e.
FN (x ) =
(11.A)
N −1
1 X
N
Dn (x ),
n =0
where Dn stands for the n-th Dirichlet kernel. Show that
FN (x ) =
(11.B)
sin(Nx/2)2
.
N sin(x/2)2
Exercise 12. Show that
Z
∞
(12.A)
sin(x )
x
0
dx =
π
2
.
Heisenberg’s Uncertainty Principle
Exercise 13. Let ψ ∈ S (R) such that
Z
R∞
−∞
|ψ(x )| dx = 1. Show that
∞
! Z
2
∞
2
x |ψ(x )| dx
(13.A)
−∞
!
2
2
ξ |ψ̂(ξ )| dx ≥
−∞
where equality holds iff ψ(x ) = Ae−Bx , for some B > 0, A =
2
2
1
16π 2
√
2B/π.
,
Exercise 14. The aim of this exercise is to give an alternative proof of (13.A).
Let L : S (R) → S (R) be the Hermite operator
d2 f
(x ) + x 2 f (x )
dx 2
Lf (x ) := −
(14.A)
and let h·, ·i be the L 2 (R) inner product on S (R) given by
∞
Z
hf, gi :=
(14.B)
f (x )g(x ) dx.
−∞
i. Show that Heisenberg’s Uncertainty Principle implies that
hLf, f i ≥ hf, f i.
(14.C)
ii. Let A, A∗ be the operators
df
(x ) + xf (x ),
dx
df
A∗ f (x ) := −
(x ) + xf (x ).
dx
Af (x ) :=
(14.D)
(14.E)
Prove that
a. hAf, gi = hf, A∗ gi.
b. hAf, Agi = hA∗ Af, gi.
c. A∗ A = L − I.
Give an alternative proof of (ii).
iii. For t ∈ R, let At , At∗ be the operators
df
(x ) + txf (x ),
dx
df
At∗ f (x ) := −
(x ) + txf (x ).
dx
At f (x ) :=
(14.F)
(14.G)
Given that hAt∗ At f, f i ≥ 0 show that
Z
∞
! Z
2
∞
2
x |f (x )| dx
(14.H)
−∞
−∞
df
dx
2
!
dx ≥
1
4
.
Fourier Transform and Fourier Series
Exercise 15. Let f : R → R such that fˆ is of moderate decrease and supp f ⊂ [−M, M ].
i. Fix L > 0 such that L/2 > M and show that
f (x ) =
(15.A)
X
an (L )e2π inx/L ,
n ∈Z
uniformly in x, where an (L ) = (1/L )fˆ (n/L ). If we let δ = 1/L we get equivalently
f (x ) = δ
(15.B)
X
fˆ (nδ )e2π inδx .
n ∈Z
ii. Let F : R → C be a continuous function of moderate decrease. Show that
Z
∞
(15.C)
−∞
F (ξ ) dξ = lim+ δ
δ →0
X
F (nδ ).
n ∈Z
iii. Conclude that
(15.D)
f (x ) =
Z
∞
fˆ (ξ )e2π ixξ dξ.
−∞
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References
[1] E. Stein, R. Shakarchi. Fourier Analysis. Princeton University Press, 2003.
[2] A. Giannopoulos. Ανάλυση Fourier και Ολοκλήρωµα Lebesgue (Greek). Lecture Notes,
University of Athens, Department of Mathematics, 2012.
Pavlos Tsatsoulis
University of Warwick
Coventry, UK
p.tsatsoulis@warwick.ac.uk
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