Prof. Girardi
Properties of the Fourier Coefficients
Definitions and Notation
Let f ∈ L1 (T, C). Or equivalently, using the natural identification of T with [−π, π),
let f ∈ L0 (R, C) be a 2π-periodic function such that f |[−π,π] ∈ L1 ([−π, π] , C). Then
Z
kf kL1 (T) =
1
2π
|f (θ)| dµT (θ) =
T
where λ is Lebesgue measure on R and µT =
Z
−inθ
f (θ)e
fb(n) =
λ
.
2π
Z
π
|f (t)| dλ (t) =
−π
1
2π
kf kL1 ([−π,π])
The nth Fourier coefficients of f , for n ∈ Z, is
dθ =
1
2π
Z
T
π
f (t)e−int dt.
−π
Note that for f ∈ L1 (T, C), we can view fb: Z → C. Next, consider a function g of the form
g: D → C
where D is either T or R or Z. Let a ∈ D. Then we can define the functions
τa g, g ◦ , g : D → C
by, for t ∈ D,
(τa g) (t) := g(t − a)
g ◦ (t) := g(−t)
g(t) := g(t) .
Properties of the Fourier Coefficients for f ∈ L1 (T, C)
FC1
b f (n) ≤ kf kL1 (T)
FC2
f[
∗ g = fb gb
∀n ∈ Z
FC3a (τy f )b(·) = e−i(·)y fb(·)
FC3b ein(·) f (·) b = τn fb
◦
FC4 (f ◦ )b = fb
∀n ∈ Z
FC5
Riemann-Lebesgue Lemma
∀y ∈ R
FC6
lim|n|→∞ fb(n) = 0
h
i
(f ◦ ) b = fb
thus if (f ◦ ) = f then fb(n) ∈ R.
FC7
(f 0 )b(·) = i (·) fb(·)
if f is absolutely continuous (see below).
Properties FC3 and FC7 are true beauties. Think what they are saying.
2014 November 14
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Prof. Girardi
Properties of the Fourier Coefficients
Absolutely continuous functions
The proofs of the states of this section can be found in the below references.
Definition. A function f : [a, b] → C is called absolutely continuous provided for all ε > 0 there
is a δ > 0 such that for all choices of disjoint subintervals Ii = (ai , bi ) of [a, b], i = 1, ..., n, with
n
X
(bi − ai ) < δ,
i=1
one has
n
X
|f (bi ) − f (ai )| < ε.
i=1
The collection of all absolutely continuous functions from [a, b] into C is denoted by AC ([a, b]).
Definition. A function f : [a, b] → C is called Lipschitz continuous provided there exists a C ∈ R
so that, for all s, t ∈ [a, b],
|f (s) − f (t)| ≤ C |s − t| .
The collection of all Lipschitz continuous functions from [a, b] into C is denoted by Lip ([a, b]).
Remark. For a compact interval [a, b], we have the inclusions
C 1 ([a, b]) ⊆ Lip ([a, b]) ⊆ AC([a, b]) ⊆ C([a, b])
and none of the reverse inclusion holds.
Here is a characterization of absolutely continuous functions.
Theorem. Let f : [a, b] → C. Then f is absolutely continuous if and only if there exists a
function h in L1 ([a, b], C) such that
Z x
h(t)dt
f (x) = f (a) +
a
for all x ∈ [a, b]. In particular, f is differentiable almost everywhere with derivative f 0 = h.
In particular, the integration by parts formula also holds for absolutely continuous functions.
Theorem. Let f and g be absolutely continuous functions on an interval [a, b]. Then
Z b
Z b
0
f (t) g (t) dt = f (b) g(b) − f (a) g(a) −
f 0 (t) g (t) dt.
a
a
References
[1]
[2]
[3]
Gerald B. Folland, Fourier analysis and its applications, The Wadsworth & Brooks/Cole Mathematics Series,
Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York, 1975, A modern
treatment of the theory of functions of a real variable, Third printing, Graduate Texts in Mathematics, No.
25.
Frank Jones, Lebesgue integration on Euclidean space, Jones and Bartlett Publishers, Boston, MA, 1993
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