Extra Problems to MAT2400, Chapter 7 – Spring 2016
Problem 1. (The first three parts of this problem are almost identical to
Problem 7.2.3 in the book, but they form a necessary background for the rest of
the problem.) We shall first show
that if f : [−π, π] → R is a real valued function
P∞
in D, then the Fourier series n=−∞ αn en can be turned into a sine/cosineseries of the form (7.2.2).
a) Show that if αn = cn + idn are Fourier coefficients of f , then α−n = αn =
cn − idn .
Rπ
Rπ
1
1
b) Show that cn = 2π
f (x) cos(nx) dx and dn = − 2π
f (x) sin(nx) dx.
−π
−π
c) Show that the Fourier series can be written
α0 +
∞
X
2cn cos(nx) − 2dn sin(nx)
n=1
To get rid of the factors 2 and the minus in the sine part, we shall from now on
write
Z
1 π
f (x) cos(nx) dx, n = 0, 1, 2, 3, ...
an =
π −π
and
1
bn =
π
Z
π
f (x) sin(nx) dx,
n = 1, 2, 3, ...
−π
d) Show that the Fourier series now can be written
∞
a0 X
+
an cos(nx) + bn sin(nx)
2
n=0
A function f : [−π, π] → R is called even if f (−x) = f (x) for all x ∈ [−π, π]
and it is called odd if f (−x) = −f (x) for all x ∈ [−π, π].
d) Show that if f is even, bn = 0 for all n = 1, 2, 3, . . ., and that if f is odd,
an = 0 for n = 0, 1, 2, . . .. In the first case, we get a cosine series
∞
a0 X
+
an cos(nx)
2
n=0
and in the second case a sine series
∞
X
bn sin(nx)
n=0
1
e) Show that
π
4
|x| = −
2
π
cos 3x cos 5x
+
+ ...
cos x +
32
52
for all x ∈ [−π, π].
f) Show that
| sin x| =
∞
2
4 X cos(2nx)
−
π π n=1 4n2 − 1
for all x ∈ [−π, π]. (Hint: Show first that sin[(n + 1)x] − sin[(n − 1)x] =
2 sin x cos nx.)
g) Show that
sin x +






sin 3x sin 5x
+
+ ... =

3
5




π
4
if x ∈ (0, π)
0
if x = 0
− π4
if x ∈ (−π, 0)
h) Show that if a ∈ R, a 6= 0, then
e
ax
eaπ − e−aπ
=
π
∞
X (−1)n
1
+
a cos nx − n sin nx
2
2
2a n=1 n + a
!
for all x ∈ (−π, π).
i) Show that for x ∈ (0, 2π),
sin 2x sin 3x
+
+ ...
x = π − 2 sin x +
2
3
(Warning: Note that the interval is not the usual [−π, π]. This has to be
taken into account.)
Problem 2. Let f : [−π, π] → R be a real valued function function in D as in
Problem 1 and assume that
∞
a0 X
+
an cos(nx) + bn sin(nx)
2
n=0
is the real Fourier series of f .
a) Show that fe (x) =
f (x)+f (−x)
2
is an even function with real Fourier series
∞
a0 X
+
an cos(nx)
2
n=0
2
and that fo (x) =
f (x)−f (−x)
2
is an odd function with real Fourier series
∞
X
bn sin(nx)
n=0
b) Show that the Fourier series of g(x) = f (π − x) is
∞
a0 X
+
(−1)n an cos(nx) + bn sin(nx)
2
n=0
Problem 3. We shall prove the following statement:
Assume that f ∈ D has Fourier coefficients αn =
are positive constants c, γ ∈ R+ such that
1
2π
Rπ
−π
f (x)e−inx dx. If there
|f (x) − f (y)| ≤ c|x − y|γ
for all x, y ∈ [−π, π], then
|αn | ≤
c π γ
2 n
for all n ∈ Z.
Explain the following calculations and show that they prove the statement.
αn =
=−
Hence
1
2π
Z
1
2π
Z
π
f (x)e−inx dx =
−π
π
f (t +
−π
1
2π
Z
π
π− n
f (t +
π
−π− n
1
π −int
)e
dt = −
n
2π
Z
π −in(t+ π )
n dt
)e
n
π
f (x +
−π
π −inx
)e
dx
n
Z π
Z π
1
π
1
−inx
f (x)e
dx −
f (x + )e−inx dx|
|αn | = |
4π −π
4π −π
n
Z π
Z π γ
1
π
1
π
c π γ
≤
dx =
|f (x) − f (x + )| dx ≤
c
4π −π
n
4π −π
n
2 n
This result connects the “smoothness” of f (the larger γ is, the smoother f
is) to the decay of the Fourier coefficients: Roughly speaking, the smoother the
function is, the faster the Fourier coefficients decay (recall that by the RiemannLebesgue Lemma, |αn | → 0). This is an important theme in Fourier analysis.
3