R
f (t)e
i!t
T
1
fˆ(!) = p
2⇡
Z
AF
The Fourier transform
DR
Fabian Cádiz and Rodrigo F. Cádiz
dt
T
AF
DR
Fabian Cadiz
Ecole Polytechnique
fabiancadiz.com
Rodrigo F. Cádiz
Pontificia Universidad Católica de Chile
rodrigocadiz.com
Copyright c 2019 Fabian Cadiz and Rodrigo F. Cadiz
T
Contents
Basic functions and spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1
Introduction
1.2
Basic functions
1.2.1
The rectangular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2
The Heaviside unit step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3
The sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4
The triangular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5
The cardinal sine (Sinc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.6
The Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.7
The Cauchy-Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.8
The decaying exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.9
The complex exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3
Basic operations with functions
1.3.1
Dilation and contraction of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2
Translation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4
Integration
1.4.1
Limitations of the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2
The Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.3
Integration over a subset A ∈ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
10
10
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13
13
1.4.4
Convolution between two functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5
Generalized functions: distributions
17
1.6
Functional spaces and convergences
18
1.6.1
The norm of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.2
The space of bounded functions L∞ (R; C) . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.3
The space of integrable functions L1 (R; C) . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.4
The space of square integrable functions L2 (R; C) . . . . . . . . . . . . . . . . 22
1.6.5
Completeness of Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.6
Inclusion properties of L1 and L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7
Different types of convergence
1.7.1
Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.2
Convergence almost everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.3
Uniform convergence or L∞ -convergence . . . . . . . . . . . . . . . . . . . . . . 24
1.7.4
Convergence in mean or L1 −convergence . . . . . . . . . . . . . . . . . . . . . 25
1.7.5
Convergence in quadratic mean or L2 −convergence . . . . . . . . . . . . 25
2
Fourier transform in L1 (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1
Introduction
28
2.2
The Fourier transform in L1
28
2.2.1
The Fourier transform is a continuous application F : L1 (R; C) 7→ C0 (R; C)
29
2.3
Some basic properties properties of the Fourier transform
2.3.1
Fourier transform and translation/modulation . . . . . . . . . . . . . . . . . . . . 36
2.3.2
Fourier transform and dilation/contraction . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3
Fourier transform and differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.4
The Fourier transform exchanges smoothness and decay rate . . . . . . 41
2.3.5
Fourier transform of a convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4
The inverse Fourier transform
43
2.5
Fourier transform of a product of L1 functions
46
2.5.1
Fourier transform and convolution (summary) . . . . . . . . . . . . . . . . . . . . 47
2.6
The transfer formula
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24
35
47
2.7
The Plancherel-Parseval theorem
48
2.8
Summary
50
3
The Schwartz space S(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1
The Schwartz space S
54
3.2
The Fourier transform is a linear bijection S(R; C) 7→ S(R; C)
56
3.3
Properties of the Fourier transform in S(R; C)
57
3.3.1
Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2
Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.3
Multiplication and convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.4
Plancherel-Parseval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4
Summary
4
Fourier transform in L2 (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1
Introduction
4.1.1
An intuitive definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2
Fourier transform in L2 (R; C)
62
4.3
Properties of the Fourier transform in L2 (R; C)
67
4.3.1
Integral definition of the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 67
4.4
Fourier transform and differentiation in L2
69
4.5
Fourier transform and convolution
70
4.6
Fourier transform and multiplication
70
4.7
Heisenberg’s inequality
70
4.7.1
Proof of the Heisenberg’s unequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5
Fourier transform of tempered distributions . . . . . . . . . . . 73
5.1
Introduction
74
5.2
Distributions
74
5.2.1
Functions with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.2
The space of test functions D(Ω; C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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58
62
5.2.3
The space of distributions D0 (Ω; C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3
Examples of distributions
5.3.1
Regular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2
The Dirac distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.3
The Cauchy’s principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.4
Restriction of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.5
Support of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.6
Convergence in D0 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4
Operations on distributions
5.4.1
Translation of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.2
Dilation/Contraction of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.3
Derivative of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.4
Multiplication by a C ∞ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5
The space of tempered distributions S 0 (R)
T
89
5.6
Linear applications of tempered distributions
90
5.6.1
Multiplication of a distribution by a tempered function . . . . . . . . . . . . 91
5.6.2
Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6.3
Differentiation and Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6.4
Solution of differential equations with distributions . . . . . . . . . . . . . . . . . 93
5.7
The Fourier transform of tempered distributions
93
5.8
The inverse Fourier transform
94
5.8.1
Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9
Compatibility with the Fourier transform in L1 and L2
96
5.10
Properties of the Fourier transform in S 0
96
76
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80
5.10.1 Fourier transform in E 0 (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.10.2 The Dirac’s comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.11
Sampling
103
5.12
Summary
105
6
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1
Introduction
108
6.2
Fourier series of periodic functions
108
6.3
Periodic distributions
113
6.4
Link between the Fourier series and the Fourier transform
115
7
Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1
Introduction
118
7.2
Laplace Transform
118
7.2.1
Link between Laplace and Fourier transforms . . . . . . . . . . . . . . . . . . . 119
7.3
Properties of the Laplace transform
7.3.1
Laplace transform and translation/modulation . . . . . . . . . . . . . . . . . . 122
7.3.2
Laplace transform and dilation/contraction . . . . . . . . . . . . . . . . . . . . 122
7.3.3
Laplace transform and differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3.4
Laplace transform and integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.5
Laplace transform of a convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4
Aymptotic behaviour
7.4.1
Behaviour at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4.2
The final value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4.3
The initial value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5
The inverse Laplace transform
127
7.6
The Laplace transform of a distribution
129
7.6.1
Laplace transform of a distribution and differentiation . . . . . . . . . . . . 130
8
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.1
Vector spaces
134
8.2
Convolution between functions
134
8.3
Convolution between distributions and functions
135
8.4
Convolution between distributions
136
8.5
Monotone convergence theorem
136
8.6
Differentiation under the integral sign
137
8.7
Dominated convergence theorem
137
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125
8.8
Fubini theorem
138
8.9
Approximation of the identity
138
8.10
The Riemann-Lebesgue theorem
138
8.10.1 Proof of the Riemann-Lebesgue theorem . . . . . . . . . . . . . . . . . . . . . . . 138
8.11
Proof of Fourier’s inversion formula in L1
140
8.12
Complex analysis
141
8.12.1 Derivative of a complex-valued function . . . . . . . . . . . . . . . . . . . . . . . 141
8.12.2 Holomorphic and analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.12.3 Cauchy-Goursat’s integral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.12.4 Cauchy’s integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.12.5 Pole of an holomorphic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.12.6 Residue of a function around a pole . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
AF
Proof of the Dirichlet-Jordan’s theorem
DR
8.13
T
8.12.7 Cauchy’s residue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
143
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1 — Basic functions and spaces
Basic functions and spaces
10
1.1
Introduction
In this section we will introduce the notation and the definition of several basic
functions and functional spaces whose properties under the Fourier transform will
be discussed in the following chapters.
1.2
1.2.1
Basic functions
The rectangular function
u(t) = 1(−1/2,1/2) =
1.2.2
1 if |t| < 1/2
0 otherwise
(1.1)
The Heaviside unit step
AF
(
T
The "rectangular" function u : R → [0, 1] represents a square pulse of unit width,
unit height and unit integral. It is given by:
(
H(t) =
DR
The Heaviside "step" function H : R → [0, 1] represents a discontinuity of unit
height at the origin. It may represent a signal that is "switched on" at t = 0:
1 if t ≥ 0
0 otherwise
(1.2)
note that its integral over R is not defined.
1.2.3
The sign
The "sign" function Sign : R → [−1, 1] is given by:




1 if t > 0
Sign(t) = −1 if t < 0



0 if t = 0
(1.3)
1.2 Basic functions
11
note that, once again, this function is not integrable over R. It can be written in
terms of the Heaviside function as:
Sign(t) = H(t) − H(−t)
1.2.4
The triangular function
The triangular function ∧ : R → [0, 1] represents a triangle of unit height with a
total unit integral:
∧(t) =
1.2.5
(1 − |t|) if |t| ≤ 1
0 otherwise
(1.4)
T
(
The cardinal sine (Sinc)



(1.5)
DR
sin(t)
if t 6= 0
Sinc(t) =
t


1 if t = 0
AF
The cardinal sine function Sinc : R → (0, 1] is defined as:
The Sinc is a continuous, infinitely differentiable function. Note that it decays to
zero at infinity as 1/|t|, and so its integral over R does not converge. This function
plays a crucial role in signal processing but also in optics since it represents the
light intensity obtained whenever a beam is diffracted by a rectangular aperture.
We will see that the Fourier transform relates the Sinc to the rectangular function
u.
1.2.6
The Gaussian
The Gaussian function G : R → (0, 1] of unit height,width and integral is defined
as:
This is a continuous, infinitely differentiable function. The Gaussian plays a crucial
role in solving the heat-diffusion equation, but also in image processing, statistics,
probability, and quantum mechanics. In contrast to the Sinc, the gaussian decays
very fast to zero, in particular faster than any inverse power of t.
Basic functions and spaces
12
1
2
G(t) = √ e−t /2
2π
1.2.7
(1.6)
The Cauchy-Lorentzian
The Cauchy-Lorentzian function L : R → (0, 1] is a singly peaked function defined
as:
1
1/2
π t2 + (1/2)2
(1.7)
T
L(t) =
1.2.8
DR
AF
which is normalized so that it has as a unit integral. It represents the solution
to the differential equation describing forced resonance, and it also describes the
spectral emission or absorption of light by atoms. In probabiliy, it describes the
Cauchy distribution.
The decaying exponential
The one-sided decaying exponential h : R → (0, 1] of unit height and unit integral
is defined as:
h(t) = e−t 1[R≥0 ]
(1.8)
and it may be used to represent the decay of a signal after turning off an excitation
at t = 0 (discharge of a capacitance in an RC circuit, decay of the population of
excited atoms in a discharge lamp). The double-sided exponential g : R 7→ (0, 1] is:
and has an integral of 2. Both exponentials are related by
h(t) = g(t)H(t)
we will see that the double-sided exponential is related to the Cauchy-Lorentzian
function via the Fourier transform.
1.3 Basic operations with functions
g(t) = e−|t|
1.2.9
13
(1.9)
The complex exponential
The complex exponential f : R → C of frequency ω0 is defined as:
eω0 = eiω0 t = cos ω0 t + i sin ω0 t
(1.10)
Basic operations with functions
1.3.1
Dilation and contraction of a function
AF
1.3
T
we see that a complex exponential has a unit modulus for every t, and so it is not
integrable over R. The set of complex exponentials {eω }ω∈R plays a major role in
Fourier analysis as they consititute the basis of Fourier decomposition.
∂a f (t) ≡ fa (t) =
DR
Starting from a function f , one can define the a−dilated (or contracted) version
of f by:
1
f (t/a)
|a|
a ∈ R/{0}
(1.11)
which represents in terms of the independent variable t, a dilated version of f for
|a| > 1, and a contracted version of f for |a| < 1. Note that the factor 1/|a| in
(1.11) assures that fa has the same integral over R as f .
1.3.2
Translation of a function
Starting from a function f , one can define the translated version of f by:
The operation τa thus shifts the "center of mass" of f by a quantity a. A function
is said to be periodic of period T if τT f = f .
1.4
Integration
We will not discuss in detail about integration theory, instead we will retain some
of its main properties and theorems without proof.
Basic functions and spaces
14
τa f (t) = f (t − a)
Limitations of the Riemann integral
The first rigurous definition of the integral of a function over an interval was
that of Riemann. This definition is, however, not good enough for many reasons
and nowadays Fourier analysis is properly stated in terms of the more general
Lebesgue integral, that can be applied to a more general set of functions than for
the Riemann integral.
Z ∞
AF
T
One of the main difficulties of restricting ourselves to the Riemann integral is that
is is strictly defined only for bounded functions and over bounded intervals, and so
not properly defined over R. One way to extend the Riemann integral of a function
fR over R is to define its value by means of the so-called "improper integral", noted
∞
−∞ and given by:
f (t)dt =
−∞
lim
Z b
a→−∞,b→∞ a
f (t)dt
DR
1.4.1
a ∈ R (1.12)
whenever the limit on the right exist. In many cases this limit does not exist and
the "improper" Riemann integral is undefined: different results may be obtained
depending on the way the real interval is filled. Take for example the Sign function
(1.3), one has
lim
Z a
a→∞ −a
Sign(t)dt = 0 6= lim
Z 2a
a→∞ −a
Sign(t)dt = lim a → ∞
a→∞
In addition, for the Riemann integral there is no applicable theorem stablishing
when is it allowed to pass a limit through an integral sign. If {fn }n≥1 is a sequence
of functions that converges to f , even if the convergence is uniform (to be defined
later in 1.7.3) one can still have:
lim
Z ∞
n→∞ −∞
fn (t)dt 6=
Z ∞
f (t)dt
−∞
or even worse, the limit f of a sequence {fn }n≥1 of Riemann-integrable functions
may not be Riemann-integrable.
The Lebesgue integral provides a good mathematical framework in which a clear set
of hypothesis may be used to determine whether or not the limit can be exchanged
with the integral sign.
1.4 Integration
The Lebesgue integral
R
From now on the symbol is used to represent the Lebesgue integral, a more
general definition of the integral than that of Riemann that can be applied to a
larger class of functions and for which important theorems may be easily stated.
It is a functional that, to every "integrable" function f , it associates a complex (or
real) number
f 7→
Z
f (t)dt C
R
For a function f : R 7→ C to be Lebesgue-integrable (or simply "integrable") over
R, it is necessary and sufficient that
Z
|f (t)|dt < ∞
(1.13)
R
AF
T
and the space of integrable functions is noted L1 (R; C). This is a vector space,
as will be discussed in section 1.6, and the Lebesgue integral is then a linear
application from L1 (R; C) to C .
The Lebesgue integral is such that the measure of a subset A of R is:
µ(A) =
DR
1.4.2
15
Z
1A (t)dt
R
where 1(A) is the function equal to 1 in A and 0 everywhere else. Of particular
importance for the construction of the integration theory of Lebesgue are the
"negligible" ensembles, that is ensembles of zero measure. Any finite or countable
union of negligible ensembles has still zero measure. A point being of zero measure,
an important result is that every finite or countable set of points is negligible in R.
This Lebesgue measure can be used to properly state under which conditions a
function f is Riemman-integrable. Indeed, f is Riemann-integrable over [a, b] if
and only if f is bounded and the set of points for which f is discontinuous has
Lebesgue measure zero.
Definition 1.4.1 — Almost everywhere. We say that the property P (t) depend-
ing on the point t is verified for almost every value of t if the set {t| P (t) is false}
has zero measure.
An important result of the theory of integration is the following: Let f be a positive
function f : R 7→ R≥0 , then:
Z
R
f (t)dt = 0 ⇐⇒ f (t) = 0 for almost every value of t
(1.14)
Basic functions and spaces
16
An immediate consequence is that if the functions f and g are equal for almost
every t, and if one of them is integrable, then the other is integrable as well and
both have the same integral. Indeed:
|
Z
f (t)dt −
Z
R
g(t)dt| ≤
Z
R
|f (t) − g(t)|dt = 0
R
If one changes the value of a funcion in a point, or more generally in a countable
set of values of t, the Lebesgue integral remains unchanged.
Integration over a subset A ∈ R
Let A be a subset of R and f a function defined over A. We note fA the function
defined over R that is equal to f for t ∈ A and 0 everywhere else. Then, f is said
to be integrable over A if fA is integrable over R, and one has
Z
f (t)dt =
A
Z
fA (t)dt
T
R
AF
In particular, the integral of a function over an ensemble of zero measure is zero.
Therefore, itRis equivalent to integrate a function f over [a, b] or ]a, b[, the result
being noted ab f (t)dt in both cases. Let us retain the two following results:
• If f is Riemann-integrable over [a, b], then f is (Lebesgue-) integrable over
[a, b] and both definitions of the integral (Riemann and Lebesgue) coincide.
DR
1.4.3
• If f is (Lebesgue-)integrable over R, one has:
Z
f (t)dt =
R
lim
Z b
a→−∞,b→∞ a
f (t)dt
∈C
Note that the left term is not an "improper" integral, but a well-defined
Lebesgue integral whose value is equal to the limit on the right if f is
integrable. A counterexample is the Sinc function defined by (1.5) which is
not Lebesgue integrable over R since
Z
R
|
sin t
|dt = ∞
t
However, it is integrable over any bounded interval and one has in particular
lim
Z a
a→∞ −a
Z
sin t
sin t
dt = π 6=
dt
t
R t
Some important theorems regarding the Lebesgue integral are stated in the Appendix: the dominated convergence theorem 8.7.1, differentiation under the integral
sign 8.6.1 and Fubini’s theorem 8.14.
1.5 Generalized functions: distributions
1.4.4
17
Convolution between two functions
The convolution between two functions f and g is the function h defined by:
h(t) = (f ∗ g)(t) =
Z
f (u)g(t − u)du =
R
Z
f (t − u)g(u)du
(1.15)
R
AF
T
Naturally, for the convolution to be well defined the integrals in (1.15) needs to
be finite. We will see later under which precise conditions one can define the
convolution between f and g.
DR
Figure 1.1: For every t, the convolution between f and g equals the area under
the product of f and τt g, represented here in yellow for different values of t.
The convolution plays a significant role in signal processing, in solving differential
equations and in Fourier analysis. It will be shown that indeed the Fourier
transform exchanges convolution and multiplication, and that the response of any
linear system can be characterized by a convolution.
1.5
Generalized functions: distributions
Every physical measurement has a finite uncertainty, either limited by the experimental apparatus or by nature itself (Heisenberg’s uncertainty principle). In
consequence the precise, exact value of a function f at a given value of the coordinate t is never trully accessible experimentally. Instead, if f represents a physical
quantity, every measurement of f around the coordinate t gives a weighted average
of the form
Z
f (t0 )ϕ(t0 )dt0
(1.16)
R
where ϕ is a "smooth function" (the meaning of this is to be precised later) peaked
around t0 = t that depends on the experimental conditions. The pointwise values of
Basic functions and spaces
18
f are thus not all relevant for the integral in (1.16). For example, any discontinuity
of f is averaged out in (1.16) and so the exact values of f before and after a "jump"
are irrelevant. In this context, it is better to think of f not as a function but as
a "generalized function" or "distribution" of all the possible outcomes of the form
(1.16) for all the different weighting functions ϕ.
One of the most emblematic distributions is the "impulse" or Dirac’s δ distribution.
Conceptually it represents a sudden, instantaneous impulse at t = 0 that is zero
elsewhere. It may model, for example, an impact between two bodies which happens in a timescale much shorter than the typical evolution of their positions before
and after the collision. It may also represent a voltage pulse applied to a circuit in
a timescale much shorter than the evolution of all the relevant physical quantities
of the circuit. It is impossible to properly define the value of δ for t = 0. It is indeed inadequate to model δ by a function. It is instead a well defined distribution δ.
DR
AF
T
The theory of distributions, due to the french mathematician Laurent Schwartz is
nowadays a fundamental tool of mathematics. It allowed to extend Fourier analysis
to a much larger set of objects than was originally possible when dealing with
functions only. It allowed also to generalize the concept of derivative, therefore
its importance in solving partial differential equations. A classic example is the
Heaviside function H defined in 1.2. When thinking of H as a function, it does
not have a Fourier transform and its derivative is 0 everywhere except at t = 0
where it is not defined. When thinking of H as a distribution, one obtains the
remarkable result H 0 = δ and its Fourier transform can be properly defined.
Fourier analysis of distributions will be the topic of chapter 5.1.
1.6
Functional spaces and convergences
In Fourier analysis we will focus on particular sets of complex functions that are
vector spaces over the field C (also called C-vector spaces). A set E of functions is
a C-vector space if it is closed under the opperations of addition and multiplication
by a complex number 1 :
∀ f, g ∈ E,
f +g ∈E
∀ f ∈ E, ∀ λ ∈ C,
λf ∈ E
In the context of functional spaces, every function can be though as a "vector": they
represent a given "point" in a space E. As an example, the Fourier series consists
of decomposing a given function f in terms of a particular basis of functions ei :
1
In addition, for E to be a vector space, the sum operation and scalar multiplication must
satisfy a set of axioms that can be found in the appendix 8.1.
1.6 Functional spaces and convergences
f=
X
19
ci ei
i
where ci are complex numbers representing the projection of f into the basis
element ei . The fact that functions can be manipulated as vectors will allow us
to use several theorems and tools of analysis of finite vector spaces. For example,
given a function f , we can define its "size" (or its "length") just like one can for a
vector in the complex plane. This allows to also define the distance between two
functions, so that geometrical concepts can be applied to functional spaces too.
Finally, each vector space will have its own notion of convergence, a remarkable
difference from finite vector spaces in which there is only one possible notion of
convergence.
In this section, we will define the norm and distance between functions in the most
relevant functional spaces that will be treated in the following chapters.
T
Examples of functional vector spaces
DR
AF
• The set C(R; C) of continuous functions over R is a vector space.
• The set C0 (R; C) of continuous functions over R that vanish at infinity is a
vector space.
• The set Cb (R; C) of continuous and bounded functions over R is a vector
space. In addition C0 (R; C) ⊂ Cb (R; C)
• The set C k (R; C) of continuous functions over R that have continuous first k
derivatives is a vector space. In addition C n (R; C) ⊂ C m (R; C) for m < n.
• A function f is p-integrable over R , with p ∈ [1, ∞[ if
Z
|f (t)|p < ∞
R
The space of p-integrable functions Lp (R; C) with p ≥ 1 is a vector space.
1.6.1
The norm of a function
Let E be a C-vector space of functions. Let k k : E → R≥0 be an application that
verifies:
• (1) kf k = 0 if and only if f = 0.
• (2) ∀ f ∈ E and ∀ λ ∈ C, one has kλf k = |λ|kf k
• (3) ∀ f, g ∈ E, one has the triangular inequality kf + gk ≤ kf k + kgk
we say then that (E, k k) is a normed vector space, and k k is called norm.
Basic functions and spaces
20
Definition 1.6.1 — Distance between two functions. If (E, k k) is a normed
vector space, then it is possible to define the distance d between f, g ∈ E as the
norm of the element f − g
d(f, g) := kf − gk
(1.17)
Then, in a given normed vector space of functions we can define the distance
between two functions in complete analogy to the distances between points in
RN . From the definition 1.17, it is clear that d(f, g) = 0 if and only if f = g,
and that the triangular inequality holds for the distances between functions
d(f, g) ≤ d(f, h) + d(h, g). Once the distance between functions is defined, it is
possible to associate a topology to the space E: one can define open and closed
subsets, spheres, balls, and crucially, the concept of convergence of a sequence.
Definition 1.6.2 — Convergence. Let {fn }n∈N be a sequence of elements of the
vector space (E, k kE ). We say that fn converges to f ∈ E if:
T
lim kfn − f kE = 0
n→∞
AF
and we write
E
fn −
→f
The space of bounded functions L∞ (R; C)
DR
1.6.2
Let us consider the vector space of bounded complex functions defined over R,
denoted L∞ (R; C). This is indeed a vector space; a linear combination of two
bounded functions is bounded, and mutliplication of a bounded function by a
complex scalar remains bounded. The norm of a bounded function in L∞ (R; C),
denoted as kf k∞ , is defined as
kf k∞ := sup |f (t)|
t∈R
∈ R≥0
(1.18)
it is easy to verify that k k∞ satisfy the properties of a norm given in 1.6.1. In this
space, the distance between two bounded functions f and g is therefore given by:
kf − gk∞ = sup |f (t) − g(t)|
t∈R
so that the distance is zero if, and only if, f = g for every value of t ∈ R. We
will see later that in other vector spaces, two functions can be at "zero distance"
without being equal pointwise, they can still differ at some particular values of t
provided the norm of the difference remains zero.
1.6 Functional spaces and convergences
21
Remark
The space of continous functions that vanish at infinity C0 (R; C) is a subset of
L∞ (R; C).
Example 1.1 — Norms and distances between bounded functions. The Sinc
function defined by (1.5) belongs to L∞ (R; C) and its norm in this space is given
by:
kSinck∞ = sup |Sinc(t)| = 1
t∈R
Similarly, for the rectangular function
k u k∞ = sup | u (t)| = 1
t∈R
so that both the Sinc and the rectangular function lie in the "unit" sphere of
L∞ (R; C). The distance between these two functions in L∞ (R; C) is given by:
T
k u −Sinck∞ = sup | u (t) − Sinc(t)| = 2 sin(1/2) ≈ 0.959
AF
t∈R
this way of quantifying how "near" two functions are is at the heart of functional
analysis.
The space of integrable functions L1 (R; C)
DR
1.6.3
The set of complex functions f : R → C that are integrable over R form a particular
vector space since the Fourier transform is traditionally defined in this space, as it
will be shown in chapter 2.1. We recall that a function is integrable over R if:
Z
|f (t)|dt < ∞
R
The set of integrable functions form the vector space L1 (R; C). For example, any
linear combination of two integrable functions remains an integrable function, since
for f, g ∈ L1 (R; C), we have
Z
|α1 f (t) + α2 g(t)|dt ≤ |α1 |
R
Z
R
|f (t)|dt + |α2 |
Z
|g(t)|dt < ∞ α1 , α2 ∈ C
R
It is possible to define the L1 -norm of every element of L1 (R; C) as the following
number, which is always positive or zero:
kf kL1 :=
Z
R
|f (t)|dt
∈ R≥0
(1.19)
Basic functions and spaces
22
Here we face a subtle point: two functions may be at zero distance but still differ
for some values of t. This is a problem if we want (1.19) to be a norm in the strict
sense. To solve this issue, we say that two functions f and g are equivalent in
L1 (R; C) if the norm of the difference is zero, kf − gkL1 = 0, and this is denoted
by f ∼ g. Two equivalent functions f and g are therefore identical for (at least)
almost every value of t ∈ R. For example, the "rectangular" function defined as:
(
u(t) =
1 if |t| < 1/2
0 otherwise
is equivalent in L1 (R; C) to
(
1 if |t| ≤ 1/2
0 otherwise
u∗ (t) =
The space of square integrable functions L2 (R; C)
DR
1.6.4
AF
T
since
both differ only at t = ±1/2. Their distance is zero under the L1 -norm
R
( R | u (t) − u∗ (t)|dt = 0). From now on, we will make no distinction between
two equivalent functions, so that everytime we consider an integrable function
f ∈ L1 (R; C), we are really considering f and the set of all equivalent functions
of f as being the same element. In any case, the pointwise value of any physical
quantity is never trully accessible experimentally, all that really matters is their
behaviour inside integrals as discussed in 1.5.
Another particularly useful functional space is that of square integrable functions
for which
Z
|f (t)|2 dt < ∞
R
For example in quantum mechanics, the state of a particle is represented by a
complex function ψ whose squared modulus at x represent the probability density
of finding the particle around x, so that:
Z
|ψ(x)|2 dx = 1
R
As we will see in 4.1, this space has the remarkable property of being invariant
under the Fourier transform: the fourier transform of any element L2 (R; C) is also
a square integrable function.
In L2 (R; C) it is possible to define the inner product between two elements f and
g of L2 (R; C) via
(f, g) =
Z
R
f (t)∗ g(t)dt
1.6 Functional spaces and convergences
23
so that the L2 -norm of an element f ∈ L2 (R; C) is
kf kL2 =
q
Z
2
1/2
|f (t)| dt
(f, f ) =
<∞
R
in complete analogy with the inner product and the norm of a vector in a finite
vector space. Once again, two elements f and f ∗ are equivalent in L2 (R; C) if they
only differ in a finite or countable set of values, so that kf − f ∗ kL2 = 0. We will
also retain the Cauchy-Schwarz inequality:
|(f, g)| ≤ kf kL2 kgkL2
1.6.5
(1.20)
Completeness of Lp spaces
T
One remarkable property of the normed vector spaces Lp (R; C), p ∈ [1, ∞] is that
they are Banach spaces, or in other words, they are complete under their respective
norms k kLp . This means that any Cauchy sequence of elements of Lp (R; C), that
is, any sequence such that
AF
lim kfj − fk kLp = 0
j,k→∞
1.6.6
DR
converges to an element f ∈ Lp (R; C). This completeness is of crucial importance
in the construction of the Fourier transform.
Inclusion properties of L1 and L2
• There is no inclusion relation between L1 (R; C) and L2 (R; C), for example
1
2
√ e−t ∈ L1 (R; C) ∈
/ L2 (R; C)
t
1
√
∈ L2 (R; C) ∈
/ L1 (R; C)
2
1+t
and
2
e−t ∈ L1 (R; C) ∩ L2 (R; C)
• A function f ∈ L1 (R; C) that is bounded belongs to L2 (R; C). Indeed,
|f (t)| ≤ supt∈R |f (t)| → |f (t)|2 ≤ kf k∞ |f (t)| and therefore
kf kL2 =
Z
R
|f (t)|2 dt ≤ kf k∞
Z
|f (t)|dt = kf k∞ kf kL1 < ∞
R
• If f ∈ L2 (R; C) and vanishes outside a finite interval, then f ∈ L1 (R; C)
• The product h = f g between an integrable function f ∈ L1 (R; C) and a
continuous and bounded function g ∈ Cb (R; C) belongs to L1 (R; C).
Basic functions and spaces
24
1.7
Different types of convergence
A major difference between functional spaces, which are of infinite dimension, and
finite vector spaces, is that there is a variety of different notions of convergence,
according to 1.6.2. For a sequence of functions {fn }n∈N , there is more than one
way in which it can converge to a function f and it is important to specify which
one is being employed.
1.7.1
Pointwise convergence
Let {fn }n∈N be a sequence of functions fn : R 7→ C. We say that the sequence fn
converges pointwise to f if, for every t ∈ R, the sequence fn (t) converges to the
number f (t). The notation that will be used is
pointwise
fn −−−−−→ f
or, simply
∀t ∈ R
T
fn (t) → f (t)
P
AF
Similarly, the sequence S(t) = n∈N un (t) converges pointwise if the sequence of
P
partial sums N
n=1 un (t) converges to S(t).
1.7.2
DR
To determine wether a sequence or a series converges pointwise, one uses for each
value of t the usual tools related to numerical sequences.
Convergence almost everywhere
Let {fn }n∈N be a sequence of functions fn : R 7→ C. We say that fn converges to
f almost everywhere, and we write
a.e
fn −→ f
if the set of values of t for which there is no pointwise convergence is of zero
measure. Note that this notion of convergence is less restrictive than the pointwise
convergence and it is enough to assure the convergence of integrals (for example
if the fn are bounded by an integrable function, see the dominated convergence
theorem 8.7.1).
1.7.3
Uniform convergence or L∞ -convergence
In the space L∞ (R; C) of bounded functions, a sequence {fn }n∈N ∈ L∞ (R; C)
converges to f in L∞ (R; C) if
lim kfn − f k∞ = lim sup |fn (t) − f (t)| = 0
n→∞
n→∞ t∈R
which is also called uniform convergence and we write
1.7 Different types of convergence
25
L∞
fn −−→ f
an important subspace of L∞ (R; C) is that of the continuous and bounded functions
Cb (R; C). In this subspace one can also define the uniform convergence in terms
of the L∞ −norm restricted to Cb (R; C).
1.7.4
Convergence in mean or L1 −convergence
Let {fn }n∈N ∈ L1 (R; C) be a sequence of integrable functions. The sequence
converges to an integrable function f if:
Z
lim kfn − f kL1 = lim
n→∞
n→∞ R
|fn (t) − f (t)|dt = 0
this is called convergence in mean (or L1 -convergence) and we write
L1
AF
Convergence in quadratic mean or L2 −convergence
Let {fn }n∈N ∈ L2 (R; C) be a sequence of square integrable functions. The sequence
converges to a square- integrable function f if:
lim
lim kfn − f kL2 = n→∞
n→∞
DR
1.7.5
T
fn −→ f
Z
1/2
|fn (t) − f (t)|2 dt
=0
R
this is called convergence in quadratic mean (or L2 -convergence) and we write
L2
fn −→ f
T
AF
DR
DR
AF
T
2 — Fourier transform in L1(R)
Fourier transform in L1 (R)
28
2.1
Introduction
In this chapter, the traditional definition of the Fourier transform will be given
in terms of an integral over R. The Fourier transform states that any integrable
function can be thought as a superposition of an infinite number of complex
exponentials. It is important to note that this definition, however, is restricted to
the relativelly small class of integrable functions (the functional space L1 (R; C)).
The main properties of the Fourier transform in L1 (R; C) will be discussed and we
will extend the definition of the Fourier transform to a larger family of functions
(and distributions) in the following chapters.
The Fourier transform in L1
Definition 2.2.1 — Fourier transform in L1 (R; C). To every integrable function
Remarks
(2.1)
AF
1 Z
ˆ
f (ω) = √
f (t)e−iωt dt
2π R
T
f ∈ L1 (R; C), the Fourier transform F of f is the function fˆ = F {f } defined
over R by the integral:
• The integral in (2.1) exists if and only if f ∈ L1 (R; C).
DR
2.2
√
• The normalization factor 1/ 2π can be chosen arbitrarily. The convention
used in (2.1) has some advantages that will be discussed later.
• Whereas the function f does not need to be defined for every value of
t ∈ R (we consider f and all of its equivalent functions as the same element
of L1 ), this is not the case for its Fourier transform, which is uniquely
defined for every value of ω ∈ R by the value of the integral (2.1).
• For every f ∈ L1 (R; C), its Fourier transform fˆ is a continuous, bounded
function (fˆ ∈ Cb (R; C)). Indeed, fˆ is bounded by the L1 -norm of f ,
|fˆ(ω)| ≤ √12π kf kL1 for every ω ∈ R. In addition, fˆ falls to zero when
|ω| → 0. This is the Riemann-Lebesgue theorem whose demonstration is
found in 8.10.1.
• We see then that the Fourier transform F : f 7→ fˆ defined by (2.1) is a
linear map from L1 (R; C) to C0 (R; C), the space of continuous functions
that tend to zero at infinity. Since an element of C0 (R; C) does not
necessarily belong to L1 (R; C), it is not possible in general to define the
Fourier transform of fˆ. We already face a big limitation of the Fourier
transform restricted to integrable functions.
2.2 The Fourier transform in L1
The Fourier transform is a continuous application F : L1 (R; C) 7→ C0 (R; C)
Since every continuous function φ that decays to zero at infinity (φ ∈ C0 (R; C)) is
bounded, we can define its C0 -norm via the L∞ -norm:
kφkC0 = kφk∞ = sup |φ(t)|
t∈R
then, the Fourier transform is a continous application F : L1 (R; C) 7→ C0 (R; C).
L1
This means that if (fn )n∈N is a sequence of elements of L1 (R; C) such that fn −→ f ,
L∞
then fˆn −−→ fˆ ∈ C0 (R, C). Indeed, we have
1
|fn (ω) − f (ω)| = |fn − f (ω)| ≤ √ kfn − f kL1 ∀ω ∈ R
2π
V
V
V
so that
1
sup |fn (ω) − f (ω)| ≤ √ kfn − f kL1
2π
ω∈R
V
AF
and since limn→∞ kfn − f kL1 = 0
V
∀n ∈ N
T
V
V
V
V
lim sup |fn (ω) − f (ω)| = n→∞
lim kfn − f k∞ = 0
n→∞
ω∈R
which proves that fˆn converges uniformly to fˆ in C0 (R; C).
DR
2.2.1
29
Example 2.1 — Fourier transform of the rectangular function. Let us consider
the rectangular function ∂a u = ua of width a > 0, defined as ua (t) = a1 u (t/a) for
a > 0 and where u(t) = 1(−1/2,1/2) (5.11). Clearly, ua ∈ L1 (R; C), since
k ua kL1
1Z
dt = 1 < ∞
=
a (−a/2,a/2)
and so it has a Fourier transform. Note that the precise value at the discontinuities
t = ± a2 is irrelevant, we could have chosen ua (±a/2) = 1 (or any other value) and
the resulting function will be equivalent to ua in L1 (R; C) and will have the same
Fourier transform. Let us calculate is Fourier transform, for ω 6= 0:
1 Z a/2 e−iωt
2 Z a/2
cos(ωt)dt
ua (ω) = √
dt = √
2π −a/2 a
2πa 0
V
Fourier transform in L1 (R)
30
V
ua (ω) = √
)
1 sin( ωa
2 sin(ωa/2)
2
=√
ωa
ω
2πa
2π
2
and
V
ua (0) = √
1 Z a/2
1
dt = √
2πa −a/2
2π
recaling the definition of the Sinc function given by (1.5), we see that the Fourier
transform of a rectangular function is a Sinc function of the form:
1
ωa
1 2
ua (ω) = √ Sinc( ) = √
Sinc2/a (ω)
2
2π
2π a
DR
Remarks
AF
T
V
• Whereas ua has discontinuities, its Fourier transform is continuous and
bounded for every ω ∈ R (as expected from the Riemann-Lebesgue theorem),
indeed
1
1
ωa
sup | √ Sinc( )| = √ ≤ k ua kL1 = 1
2
2π
2π
ω∈R
• The Fourier transform of ua indeed tends to zero when |ω| → ∞. However,
it is not integrable, since the integral over R of | sin t|/t diverges. We can’t
define the Fourier transform of a Sinc with Eq.(2.1).
• The larger the width ∆t = a of the rectangular function, the more concentrated its Fourier transform will be around ω = 0, since its width ∆ω is
inversely proportional to a. This is a general result that will be discussed
later (uncertainty principle).
Example 2.2 — Approximation of the Dirac δ distribution. Let us consider the
following sequence of rectangular functions (fn = u1/n )n∈N . As n increases, the
width of fn = u1/n decreases as 1/n whereas its height increases as n. The function
fn gets more and more concentrated near t = 0 as n increases, and it is often said
that (u1/n )n∈N is an approximation of the Dirac δ distribution (we will see the real
meaning of this "approximation" in the following chapters).
2.2 The Fourier transform in L1
31
For every continuous and bounded function h, one has
lim
Z
n→∞ R
u1/n (t)h(t) = lim
Z
n→∞ [− 1 , 1 ]
2n 2n
h(t)dt = h(0)
so that u1/n concentrates all the weight of the integral at the origin when n → ∞,
which is essentially the concept of a δ Dirac distribution. However, it would
be incorrect to assume
from this that the sequence (u1/n )n∈N converges to a δ
R
"function" such that R δ(t)h(t)dt = h(0) since
lim u1/n (t) = 0
T
∀ t ∈ R/{0},
n→∞
DR
AF
so that (u1/n )n∈N converges pointwise to a function equivalent to 0. The key
point here is that (u1/n )n∈N does not converge to 0 under the L1 -norm since
limn→∞ k u1/n kL1 = 1 6= 0. In conclusion, as long as we are interested in the
properties of the sequence (fn = u1/n )n∈N when employed inside an integral, then
it does "behave" like a δ distribution when n → ∞.
Note also that for every n ∈ N the function u1/n is integrable (of integral
R
R u1/n (t)dt = 1) and so it has a Fourier transform given by fn (ω) = u1/n (ω) =
√1 Sinc( ω ), where the width of this Sinc function increases with n. Note that
2n
2π
∀ ω ∈ R,
V
V
1
lim u1/n (ω) = √
n→∞
2π
V
We will see later that the Fourier transform
of a δ (to be defined properly later) is
√
indeed,in a certain way, equal to 1/ 2π. Once again, the pointwise convergence
of the sequence (u1/n )n∈N is of no interest since:
V
limn→∞ u1/n (ω) = 0
Example 2.3 — Fourier transform of a one-side decaying exponential. Let us
consider the following function f (t) = e−at 1[R≥0 ] for a > 0.
it is easy to verify that f ∈ L1 (R; C) since
kf kL1 =
Z ∞
0
|e−at |dt =
1
a
Fourier transform in L1 (R)
32
Let us calculate its Fourier transform
Remarks
AF
T
1 Z ∞ −at −iωt
1
1
ˆ
f (ω) = √
e e
dt = √
2π 0
2π a + iω
• We see that fˆ is continuous and it is easy to check that is bounded. Indeed
DR
1
1
1
1
|fˆ(ω)| = √ √ 2
= √ kf kL1
≤√
2
2π a + ω
2πa
2π
∀ω ∈ R
and tends to zero at infinity
lim |fˆ(ω)| = 0
|ω|→∞
in agreement with the Riemann-Lebesgue theorem.
√
• Again, fˆ is not integrable, since the integral over R of 1/ ω 2 + a2 diverges,
fˆ does not have a Fourier transform according to (2.1).
Example 2.4 — Fourier transform of a double-sided exponential. Consider now
the function f (t) = e−a|t| ∈ L1 (R; C) for a > 0, which is a double-sided decaying
exponential.
Its Fourier transform is given by
Z ∞
Z 0
1 Z −iωt −a|t|
1
−iωt −at
−iωt at
ˆ
e
e dt +
e
e dt
f (ω) = √
e
e
dt = √
−∞
2π R
2π 0
2.2 The Fourier transform in L1
(
1
1
fˆ(ω) = √
−
e−(a+iω)t
a + iω
2π
∞
0
33
1
+
e(a−iω)t
a − iω
0
−∞
)
1
=√
2π
1
1
+
a + iω a − iω
we find that the Fourier transform of a double-sided exponential is a CauchyLorentzian
Remarks
DR
AF
T
√
1
2a
fˆ(ω) = √
= 2πL2a (ω) ∈ L1 (R; C)
2
2
2π a + ω
• We see that this time the fourier transform of f (t) = e−a|t| decays as 1/ω 2 at
infinity and is integrable, which was not the case for the "one-side" decaying
exponential of example 2.3, whose abrupt discontinuity at t = 0 manifests
itself by a slowly decaying Fourier transform (as 1/ω at infinity). We will
see in sec.2.3.4 that the more smooth the function f is, the faster its Fourier
transform will decay at infinity.
• Defining the width ∆t of f as the full width at half maximum, one obtains
∆t = 2 ln 2/a. Similarly, the width of fˆ is ∆ω = 2a. We confirm again the
general result; the larger the width ∼ 1/a of the double-sided exponential,
the narrower its Fourier transform will be.
Example 2.5 — Fourier
transform of a Lorentzian. Let us consider a Lorentzian
√
of the form f (t) = 2πL2a (t) =
its Fourier transform:
2a
√1
2π a2 +t2
∈ L1 (R; C) with a > 0. Let us calculate
1 Z
2a −iωt
aZ∞
e−iωt
fˆ(ω) =
e
dt
=
dt
2π R a2 + t2
π −∞ (t + ia)(t − ia)
Fourier transform in L1 (R)
34
F (z)dz =
Z
C
F (z)dz +
[−R,R]
Z
F (z)dz
γ
AF
IR =
I
T
In order to calculate this last integral, let us consider the function F : C → C
1
such that F (z) = e−iωz (z+ia)(z−ia)
= e−iωz /(z 2 + 1). This function is holomorphic
in the complex plane except at the poles z = ±ia. Let us supose ω < 0 and let us
calculate the integral IR of F (z) over a contour C that goes along the real line,
from −R to R and then counterclockwise along a semicircle γ centered at the
origin of radius R.
If R > a, the contour C contains the pole z = +ia and therefore Cauchy’s residue
theorem (see 8.12.7) gives
DR
IR = 2πiRes(F, ia) =
π aω
e
a
Since ω < 0, along γ one has z = Reiφ for φ ∈ [0, π] and therefore |e−iωz | =
|e−iωR(cos φ+i sin φ) | = |eωR sin φ | ≤ 1. Then
|
Z
F (z)dz| ≤
γ
Z
|F (z)|dz ≤
γ
so that limR→∞
R
γ
γ
Z
1
1
πR
dz
≤
dz
=
|z 2 + 1|
R2 − 1
γ R2 − 1
F (z)dz = 0 and we obtain:
lim IR = lim
R→∞
Z
Z
R→∞ [−R,R]
F (z)dz =
Z ∞
−∞
e−iωt
π
dt = eaω
(t + ia)(t − ia)
a
Similarly, for ω > 0, we consider the integral of F over the contour C 0 in which now
the clockwise semicircle γ 0 contains the pole z = −ia and is such that |e−iωz | ≤ 1.
Again, Cauchy’s residue theorem gives
e−iωt
π
dt = 2πiRes(F, −ia) = e−aω
R→∞
a
−∞ (t + ia)(t − ia)
√
So that the Fourier transform of the Cauchy-Lorentzian f = 2πL2a is a doublesided exponential
lim IR =
Z ∞
2.3 Some basic properties properties of the Fourier transform
35
Remarks
AF
T
fˆ(ω) = e−a|ω|
DR
• In the previous example we saw that the Fourier transform of a double-sided
exponential is a Cauchy-Lorentzian. This is a remarkable result; for any
of these two functions, applying the Fourier transform twice gives back
the original function. We say then that a double-sided exponential and a
Cauchy-Lorentzian are Fourier transforms of each other.
• We will see later in sec.2.4.1 that this is quite a general property. For any
integrable function f such that fˆ is integrable, applying the Fourier transform
twice to f gives f (−t) (in other words, the Fourier transform F is "almost"
its own inverse). Since the double-sided exponential is a pair function, we
were not sensitive to the reversal of the independent variable when applying
twice the Fourier transform.
2.3
Some basic properties properties of the Fourier transform
When studying the properties of the Fourier transform, we will encounter remarkable symmetries between operations that act either over f or its Fourier transform
fˆ. For example, translating f corresponds to a multiplication of fˆ by a complex
exponential, also called modulation. Conversely, multiplying f by a complex
exponential corresponds to a translation of fˆ. The same thing occurs between
Fourier transform in L1 (R)
36
dilations and contractions: a dilation of f is equivalent to a contraction of fˆ and
vice-versa.
2.3.1
Fourier transform and translation/modulation
Theorem 2.3.1 Let f ∈ L1 (R; C) and fˆ its Fourier transform. Then, for any
a∈R
V
V
f (t − a) = e
−iaω
fˆ(ω)
eiat f (t) = fˆ(ω − a)
and
(2.2)
or, in terms of the translation operator τ
V
τa f (t) = e−iaω fˆ(ω)
and
eiat f (t) = τa fˆ(ω)
Proof
From the definition given by Eq.(2.1)
(2.3)
T
V
V
AF
1 Z −iωt
1 Z −iω(u+a)
f (t − a)(ω) = √
e
f (t − a)dt = √
e
f (u)du
2π R
2π R
V
−iωa
DR
f (t − a)(ω) = e
1 Z −iωu
√
e
f (u)du = e−iωa fˆ(ω)
2π R
Now, for the second identity
1 Z −i(ω−a)t
1 Z −iω iat
e f (t)(ω) = √
e e f (t)dt = √
e
f (t)dt = f (ω − a)
2π R
2π R
V
iat
2.3.2
V
Fourier transform and dilation/contraction
Theorem 2.3.2 Let f ∈ L1 (R; C) and fˆ its Fourier transform. For a ∈ R/ {0}
1
we define fa (t) = |a|
f (t/a). Then
V
V
fa (t) = fˆ(aω)
or, equivalently
1
f (t/a)
|a|
= fˆ(aω)
(2.4)
We see that the Fourier transform exchanges dilation with contraction, and vice
versa.
Proof
From the definition given by Eq.(2.1)
2.3 Some basic properties properties of the Fourier transform
37
1 Z −iωt f (t/a)
1 Z −iaωu
fa (t)(ω) = √
dt = √
e
e
f (u)du = f (aω)
|a|
2π R
2π R
V
V
The Fourier transform exchanges dilation with contraction, which is the reason
why the larger the width ∆t ∼ a of f , the narrower the width ∆ω ∼ 1/a of its
Fourier transform.
Fourier transform and differentiation
In general, if f ∈ L1 (R; C), its derivative is not necessarily integrable. Similarly,
its Fourier transform fˆ is not necessarily differentiable. The following theorem
will give some necessary conditions under which it is possible to define the Fourier
transform of df /dt and under which the derivative dfˆ/dω exists, and we will see
that in this case the Fourier transform exchanges multiplication by t (or ω) with
differentiation with respect to ω (or t).
T
Theorem 2.3.3 Let f ∈ L1 (R; C)
a) If tf ∈ L1 (R; C), then its Fourier transform has a continuous derivative,
f ∈ C 1 (R; C) and we have:
AF
V
V
1 Z −iωt
df (ω)
=√
e
(−it)f (t)dt = −itf (ω)
dω
2π R
V
DR
2.3.3
(2.5)
b) If f ∈ C 1 (R; C) and f 0 ∈ L1 (R; C), then:
V
df
(ω)
dt
1 Z −iωt df (t)
dt = iω fˆ(ω)
=√
e
dt
2π R
Proof
a) Let’s define the function g : R × R
g(ω, t) = e−iωt f (t) ∈ C
so that g verifies:
g(·, t) ∈ C 1 (R; C) for almost every t ∈ R
g(ω, ·) ∈ L1 (R; C) for every ω ∈ R since |g(ω, t)| = |f (t)|
we also have, for almost every t ∈ R and every ω ∈ R
(2.6)
Fourier transform in L1 (R)
38
|
∂g(ω, t)
| = | − itf (t)| ≤ |t||f (t)|
∂ω
from the theorem of differentiation under the integral sign (see 8.6.1), we obtain:
V
d 1 Z
df (ω)
1 Z ∂
=− √
(f (t)e−iωt )dt
−
f (t)e−iωt dt = − √
dω
dω 2π R
2π R ∂ω
V
df (ω)
i Z
−
=√
tf (t)e−iωt dt = itf (ω)
dω
2π R
V
V
df /dω is therefore continuous over R since it is the Fourier transform of tf ∈
L1 (R; C) (Riemann-Lebesgue theorem). It follows that
V
T
f ∈ C 1 (R; C)
AF
b) We have:
f (t) = f (0) +
Z t
0
DR
and
lim f (t) = f (0) +
t→±∞
df (u)
du
du
Z ±∞
0
df (u)
du
du
Since f 0 belongs to L1 (R; C) , f has a limit when
t → ±∞. Both limits should be
R
1
equal to zero since f ∈ L (R; C), otherwise R |f (t)|dt = +∞. Then, integrating
by parts:
V
df
(ω)
dt
iω Z
1 Z df (t) −iωt
−iωt b
=√
e
dt = lim [f (t)e
]a + √
f (t)e−iωt dt
a→∞,b→∞
2π R dt
2π R
and we obtain
V
df
(ω)
dt
= iω fˆ(ω)
We see that under the conditions of theorem 2.3.3, a multiplication by −it corresponds to a derivative in the Fourier domain ω, whereas a derivative with respect
to t corresponds to a multiplication by iω in the Fourier domain. Therefore,
multiplication by the independent variable and differentiation are exchanged by
the Fourier transform.
2.3 Some basic properties properties of the Fourier transform
39
Example 2.6 — Fourier transform of a gaussian. Let us consider the gaussian
function Ga (t) = a1 G(t/a) of width a for a > 0, where G(t) =
Ga (t) = √
1
2
2
e−t /(2a )
2πa2
2
√1 e−t /2 ,
2π
so that
∈ L1 (R; C)
the function Ga is infinitely differentiable, Ga ∈ C ∞ (R; C) and verifies
Z
Ga (t) = 1
T
R
since a gausian decreases faster at infinity than any power of t, tGa ∈ L1 (R; C)
and Ga ∈ C 1 (R; C). Then, according to theorem (2.3.3):
AF
V
V
dGa (ω)
= −itGa (ω)
dω
V
DR
Now, dGa /dt = −t/a2 Ga ∈ L1 (R; C) so that the first derivative of a gaussian is
integrable, dGa /dt ∈ L1 (R; C) and according to theorem (2.3.3) we also have:
V
dGa
(ω)
dt
V
= iωGa (ω)
combining the latter two equations,
V
1 dGa
(ω) + ωGa (ω) =
a2 dω
V
V
−it
Ga
a2
V
a
a
− i dG
= −i tG
+
dt
a2
dGa
dt
but dGa (t)/dt + t/a2 Ga (t) = 0, so that:
V
1 dGa (ω)
ωGa (ω) + 2
=0 ω∈R
a
dω
V
which is, again, the differential equation satisfied by a Gaussian function. The
solution for Ga has the form
V
V
2 ω 2 /2
Ga (ω) = Ce−a
where the constant C can be obtained by considering that
Fourier transform in L1 (R)
40
1 Z
1
√
C = Ga (0) =
Ga (t)dt = √
2π R
2π
V
We obtain the following remarkable property of a Gaussian function: its Fourier
transform is also a Gaussian
1
1
2 2
Ga (ω) = √ e−a ω /2 = G1/a (ω)
a
2π
V
T
The gaussian function G1 = G of width a = 1 is therefore an eigenfunction of the
Fourier transform:
AF
F {G(t)} = G(ω)
DR
Remark
Note that, as for the case of the rectangular function and its Fourier transform,
here the larger the width a of Ga , the narrower its Fourier transform will be, whose
width varies as 1/a.
1
Example 2.7 Let us consider the function f ∈ L (R; C) defined by f (t) =
te−at 1[R+ ] for a > 0.
This function can be written as f (t) = th(t), where h is the one-sided decaying
exponential h(t) = e−at 1[R+ ] . The Fourier transform of the latter was calculated
in example 2.3:
1
1
ĥ(ω) = √
2π a + iω
since f = th ∈ L1 (R; C) , ĥ ∈ C 1 (R; C) and by using theorem 2.3.3
2.3 Some basic properties properties of the Fourier transform
41
dĥ(ω)
1
1
fˆ(ω) = th(ω) = i
=√
dω
2π (a + iω)2
V
AF
The Fourier transform exchanges smoothness and decay rate
Corollary 2.3.4
• Let f ∈ L1 (R; C) be of class C k (R; C) (all the derivatives
α
of order α ≤ k exist and are continous). If ∀α ≤ k, ddtαf ∈ L1 (R; C), then
V
dα f
dtα
= (iω)α fˆ(ω)
DR
2.3.4
T
In contrast to the one-sided exponential h, which has a discontinuity and whose
Fourier transform is not integrable, the function f = th is continuous over R
and its Fourier transform is integrable since it decays as 1/ω 2 at infinity. This
exchange between smoothness and decay rate under the Fourier transform will be
generalized in the following corrollary of theorem 2.3.3.
and fˆ(ω) = O(1/|ω|k ) for |ω| → ∞.
Indeed, this can be demonstrated by applying k times the part (b) of
Theorem 2.3.3. Since for every α ≤ k, (iω)α fˆ is the Fourier transform
of an integrable function, the Riemann-Lebesgue theorem ensures that
∀α ≤ k, |ω|α |fˆ(ω)| → 0, and it follows that fˆ decays as 1/ω k .
Then, the more smooth the function f is (the higher the value of k), the
faster its fourier transform will decay at |ω| → ∞.
• If f is such that (1 + tk )f ∈ L1 (R; C), then fˆ is of class C k (R; C) and
dα fˆ
= (−i)α tα f ∀α ≤ k, α ∈ N
dω α
The faster the function f decays at |t| → ∞, the smoother its Fourier
transform will be. This is demonstrated by applying k times part (a)
of Theorem 2.3.3, and we deduce that all the derivatives of fˆ up to
order α ≤ k are Fourier transforms of integrable functions, and therefore
continous due to the Riemann-Lebesgue theorem.
V
Example 2.8 — Rectangular and sinc functions. We see that the rectangular
function ua is such that (1 + tk )ua ∈ L1 (R; C) for every k ∈ N. Indeed:
Fourier transform in L1 (R)
42
Z
| ua (t)(1 + tk )|dt =
R
1 Z a/2
|(1 + tk )|dt < ∞
a −a/2
Then, its Fourier transform is of class C ∞ (R; C), which is indeed the case for the
Sinc function, and we have
dα
(−i) t ua =
dω α
1
ωa
√ Sinc( )
2
2π
V
α α
!
∀α ∈ N
On the other hand, since ua is not continuous, its Fourier transform does not
decay fast enough as |ω| → ∞. Indeed, the sinc function decays as 1/ω and it is
not integrable.
V
√
2π ĥ(ω)fˆ(ω)
AF
ĝ(ω) = h ∗ f (ω) =
T
Fourier transform of a convolution
Theorem 2.3.5 Let f ∈ L1 (R; C) and h ∈ L1 (R; C). Then the convolution
g = h ∗ f ∈ L1 (R; C) and its Fourier transform is given by:
Proof
We have (Young’s inequality, see 8.2)
DR
2.3.5
kf ∗ hkL1 ≤ kf kL1 khkL1
so that the Fourier transform of g = f ∗ h exists and it is defined by
Z
1 Z −iωt
ĝ(ω) = √
e
f (t − u)h(u)du dt
R
2π R
since f (t − u)h(u) is integrable over R2 , we can apply Fubini’s theorem 8.14 and
use the following change of variables (t, u) 7→ (v = t − u, u)
1 Z −i(u+v)ω Z
e
f (v)h(u)dudv
ĝ(ω) = √
R
2π R
1
ĝ(ω) = √
2π
Z
R
e−ivω f (v)dv
Z
e−iuω h(u)du =
√
2π fˆ(ω)ĥ(ω)
R
Example 2.9 — Fourier transform of the triangle function. Consider the triangle
function ∧a (t) =
1
a
∧ (t/a) of width 2a for a > 0 defined by:
2.4 The inverse Fourier transform
(
∧a (t) =
43
1/a(1 − |t|) if |t| ≤ a
0 otherwise
(2.7)
Clearly, ∧a ∈ L1 (R; C), indeed
T
k ∧a kL1
2Z a
(1 − t/a)dt = 1 < ∞
=
a 0
AF
It is easy to check that ∧a can be written as a convolution of two rectangular
functions of width a
∧a (t) = ua ∗ ua (t)
DR
and therefore its Fourier transform will be the product of two Sinc functions, since
ua ∈ L1 (R; C)
V
∧a (ω) =
√
V
V
2πua (ω)ua (ω)
√
ca (ω) =
∧
2π
ωa
1
ωa
Sinc2
= √ Sinc2
2π
2
2
2π
2.4
The inverse Fourier transform
We saw previously that the Cauchy-Lorentzian and the double-sided exponential
are Fourier transforms of each other. This also applies to the gaussian which
Fourier transform in L1 (R)
44
happens to be its own Fourier transform. It is tempting to think that this is a
general property of the Fourier transform, however it has been made clear that
the Fourier transform f of an integrable function f does not need to be integrable.
Therefore, it is not possible in general to apply twice the operation associated to
the Fourier transform. However, for a reduced class of integrable functions, it is
possible to show that the Fourier transform of fˆ is strictly related to f .
V
V
Theorem 2.4.1 Let f ∈ L1 (R; C) such that its Fourier transform f is integrable
V
(f ∈ L1 (R; C)). Then, for almost every t ∈ R:
1 Z +iωt
√
e
f (ω)dω = f (t)
2π R
V
(2.8)
V
V
f (u) = f (−u)
AF
T
Note that if we replace t by −t, we recognize in the left side of (2.8) the Fourier
transform of fˆ, in which the role of t and ω in Eq.(2.8) has been exchanged.
This can be generalized as:
(2.9)
Remarks
DR
where u is the independent variable of f .
• If fˆ ∈ L1 (R; C) is the Fourier transform ofnf ,owe say that f is the inverse
Fourier transform of fˆ and write f = F −1 fˆ , were the operation F −1 is
defined by Eq.(2.8). Note that F −1 is almost equal to F, the only difference
is the sign of the complex exponential that appears in the integral.
• As stated in the beginning of this chapter, we consider two functions of
L1 to be equivalent when they only differ in a finite or a countable set of
values of t ∈ R. In general, the function defined by the integral in Eq.(2.8)
is equivalent to f , so not necessarily equal to f for every value of t ∈ R.
We consider, however, that both are the same element of L1 (R; C) since the
norm of their difference is zero in L1 .
• The theorem 2.4.1 is therefore valid for integrable functions f ∈ L1 (R; C)
that are equal, for almost every value of t ∈ R, to a continuous function of R
that tends to zero at |t| → ∞ (this is a consequence of the Riemann-Lebesgue
theorem, because f (−t) is the Fourier transform of fˆ(ω)).
• These functions L1 (R; C) for which fˆ ∈ L1 (R; C) are in addition bounded,
2.4 The inverse Fourier transform
45
and so both f and fˆ belong also to L2 (R; C), the functional space of square
integrable functions (see 1.6.6).
• As a consequence of Theorem 2.4.1, the Fourier transform is an injection from
L1 (R; C) to C0 (R; C) (the space of continuous functions that tends to 0 at
infinity). In other words, two distinct elements of L1 (R; C) will have different
fourier transforms. This means that if f, g are elements of L1 (R; C) such that
fˆ = ĝ, then necessarily f = g. Indeed, if fˆ = ĝ, then f − g = 0 ∈ L1 (R; C)
and the inversion theorem applies, which implies that f − g = 0 for almost
every value of t ∈ R, and so f and g represent the same element of L1 (R; C)
. The Fourier transform is, however, not surjective, since there are elements
of C0 (R; C) that are not the Fourier transform of any integrable function.
This is illustrated in the figure below.
AF
T
V
DR
• The proof of the inversion theorem is found in 8.11.
Example 2.10 — The Fourier transform of Sinc2 . In the example 2.9 it was shown
that the Fourier transform of the triangle function ∧a is a Sinc2 function, that is
1
ωa
∧a (ω) = √ Sinc2
2
2π
V
Since the triangle function is continous over R and decays to 0 at infinity, it is a
good candidate for theorem 2.4.1 and we anticipate that the Fourier transform
of
2
2 at
1
√
a Sinc should be a triangle function. Indeed, the function f (t) = 2π Sinc 2
belongs to L1 (R; C). To calculate its Fourier transform, we can for example try to
calculate it by definition:
1 Z −iωt
2 at
ˆ
f (ω) =
e
Sinc
dt
2π R
2
but this integral is not easy to calculate. Instead, we can use the inversion formula
and get
V
V
∧a (ω) = ∧a (−ω)
V
and since ∧a = f ,
Fourier transform in L1 (R)
46
V
f (ω) = ∧a (−ω) = ∧a (ω)
since the triangle function is pair. The Fourier transform of a Sinc2 is therefore a
triangle function.
1 Z −iωt 1
2 at
√
√
∧a (ω) =
Sinc
e
dt
2
2π R
2π
we write
au
1
∧a (u) ←→ √ Sinc2
2
2π
meaning that both functions are Fourier transforms of each other.
AF
T
Example 2.11 — The Fourier transform of a Sinc function. The Sinc does not
belong to L1 (R; C) and therefore we cannot apply the definition of the Fourier
transform (2.1) to this function. Indeed, if that was possible that would mean that
its Fourier transform should be continuous and at the same time equivalent to a
rectangular function in L1 (R; C), which cannot be the case. Therefore:
DR
1
ωa
1 Z
√ Sinc( )eiωt dω
ua (t) 6= √
2
2π R 2π
since the integral on the right side does not converge. The Sinc function is, however,
a square-integrable function ( Sinc ∈ L2 (R; C)) and in the next chapter we will
construct an extension of the Fourier transform to such functional space, in such a
way that its Fourier transform will be indeed equal to a rectangular function.
2.5
Fourier transform of a product of L1 functions
If f, g are elements of L1 (R; C), the product f g is not always integrable, so in
general it is not possible to define f g . But if ĝ ∈ L1 (R; C), then its inverse Fourier
transform is g which is therefore a continous and bounded function, g ∈ Cb (R; C),
which assures that the product f g ∈ L1 (R; C) (see 1.6.6).
V
On the other hand, the convolution between fˆ and ĝ is well defined since ĝ ∈
L1 (R; C) and fˆ ∈ Cb (R; C) (see 8.2). We have
(
)
Z
1 Z
1
f (t)e−iωt √
eitν ĝ(ν)dν dt
f g (ω) = √
2π R
2π R
V
2.6 The transfer formula
47
where we have written g according to the inversion formula (2.8). Since |f (t)ĝ(ν)|
is integrable over R2 , Fubini’s theorem gives:
(
)
1 Z
1 Z
√
f g (ω) = √
f (t)e−it(ω−ν) dt ĝ(ν)dν
2π R
2π R
V
one recognizes the Fourier transform of f evaluated at ω − ν
1 Z ˆ
1
f g (ω) = √
f (ω − ν)ĝ(ν)dν = √ fˆ ∗ ĝ(ω)
2π R
2π
V
2.5.1
(2.10)
Fourier transform and convolution (summary)
• If f ∈ L1 (R; C) and g ∈ L1 (R; C), then f ∗ g ∈ L1 (R; C) and
V
f ∗ g (ω) =
√
2π fˆ(ω)ĝ(ω)
(2.11)
AF
T
• If f ∈ L1 (R; C), g ∈ L1 (R; C) and ĝ ∈ L1 (R; C), then f g ∈ L1 (R; C) and
1
f g (ω) = √ fˆ ∗ ĝ(ω)
2π
V
(2.12)
2.6
DR
• We see that under certain conditions, the Fourier transform exchanges
multiplication and convolution
The transfer formula
The transfer formula plays a key role to define the Fourier transform to an extended
class of objects such as distributions.
Theorem 2.6.1 If f, g ∈ L1 (R; C), then
Z
fˆ(t)g(t)dt =
R
Z
f (t)ĝ(t)dt
(2.13)
R
Proof
Since f, g ∈ L1 (R; C), fˆ and ĝ are continuous and decay to zero at infinity
(Riemann-Lebesgue theorem). Therefore, f g ∈ L1 (R; C) and f g ∈ L1 (R; C). Both
integrals in (2.13) are well defined and:
V
V
Z
fˆ(t)g(t)dt =
R
by Fubini’s theorem
Z (
R
)
1 Z
−iut
√
f (u)e du g(t)dt
2π R
Fourier transform in L1 (R)
48
Z
fˆ(t)g(t)dt =
R
(
The Plancherel-Parseval theorem
We will see that the Fourier transform of functions that are integrable and square
integrable preserves the inner product and the norm of L2 .
Theorem 2.7.1 If f, g ∈ L1 (R; C) ∩ L2 (R; C), then their Fourier transforms are
in L2 (R; C) and one has:
(fˆ, ĝ) = (f, g);
kfˆk2L2 = kf k2L2
(2.14)
AF
T
Remark
We see then that for functions that are simultaneously integrable and squareintegrable, the Fourier transform preserves the "angles" and
√ "distances" between
functions in L2 (this is why the normalization factor 1/ 2π chosen in the definition 2.1 is particularly convenient). This will be a key property of the Fourier
transform extended to every square-integrable function (not necessarily integrable).
Proof
2
Consider the Gaussian function G(t) = √12π e−t /2 , which has an integral of 1. Let’s
form the approximation of the identity (see 8.9.1)
DR
2.7
)
Z
1 Z
−iut
g(t)e dt du = f (u)ĝ(u)du
f (u) √
R
R
2π R
Z
1
G (t) = G(t/)
we see that G ∗ f ∈ L1 (R; C) ∩ L2 (R; C) and converges to f in L1 (R; C) and in
L2 (R; C). Let’s define fj = G1/j ∗ f and gj = G1/j ∗ g. Then, fˆj is the product of
fˆ, which is bounded, and ĥ1/j , which is a Gaussian and therefore fˆj is integrable.
We conclude the same for ĝj . Being bounded and integrable, fˆj and ĝj are square
integrable and
(fj , gj ) =
Z
1 Z ∗
fj∗ (t)gj (t)dt = √
fj (t)
eiωt ĝj (ω)dω dt
R
R
2π R
Z
where we have used the inversion formula (2.8). Since |fj (t)ĝj (ω)| is integrable
over R2 , Fubini’s theorem gives
(fj , gj ) =
Z (
R
)∗
Z
1 Z −iωt ∗
√
e
fj (t)dt ĝj (ω)dω = fˆj (ω)ĝj (ω) = (fˆj , ĝj )
R
2π R
2.7 The Plancherel-Parseval theorem
49
in particular
lim kfˆj − fˆk kL2 = lim kfj − fk kL2 = 0
j,k→∞
j,k→∞
then fˆj is a Cauchy sequence in L2 , so it converges to ϕ ∈ L2 (R; C) and there
exists a sub-sequence of fˆj that converges to ϕ for almost every value of t. On the
other hand, since fj → f in L1 , we know that fˆj converges to fˆ uniformly at every
point (see 2.2.1). Therefore, fˆ = ϕ almost everywhere. Similarly, ĝj tends to ĝ in
L2 . Since the scalar product is continuous, we get
(fˆ, ĝ) = (f, g)
in particular, if f = g
T
kf k2L2 = kfˆk2L2
Example 2.12 — Rectangular function and Sinc. The rectangular function ua
AF
∈ L1 (R; C) is bounded, so it belongs to L1 (R; C) ∩ L2 (R; C). Its Fourier transform,
given by
1
ωa
ua (ω) = √ Sinc
2
2π
DR
V
indeed belongs to L2 (R; C). Applying the Plancherel-Parseval theorem 2.7.1:
k
ua k2L2
=
Z a/2
−a/2
Z ∞
dt
1
1
ωa
2
= = kua kL2 =
Sinc2
dω
2
a
a
2
−∞ 2π
V
which gives, for the particular case a = 1, the following identity:
Z ∞
−∞
sin2 ω
dω = π
ω2
Fourier transform in L1 (R)
50
Summary
• The Fourier transform F defined over integrable functions is a linear, continuous and injective application
F : L1 (R; C) 7→ C0 (R; C)
defined by:
1 Z
F {f } (ω) = fˆ(ω) = √
f (t)e−iωt dt
2π R
ω∈R
• fˆ is continuous over R, bounded (|fˆ(ω)| ≤ kf kL1 ∀ω ∈ R), and fˆ(ω) → 0
when |ω| → 0. However, fˆ is not necessarily in L1 (R; C).
• Translation
V
T
f (t − a) = e−iaω fˆ(ω)
AF
• Modulation
V
eiat f (t) = fˆ(ω − a)
• Dilation-Contraction
DR
2.8
V
V
fa (t) =
1
f (t/a)
|a|
= fˆ(aω)
a ∈ R/{0}
• Differentiation
If f ∈ C 1 (R; C) and df /dt ∈ L1 (R; C), then
V
df
(ω)
dt
= iω fˆ(ω)
• Multiplication by t
If tf ∈ L1 (R; C), then fˆ ∈ C 1 (R; C) and
V
−itf (ω) =
dfˆ(ω)
dω
• Convolution
If f, g ∈ L1 (R; C), then f ∗ g ∈ L1 (R; C) and
V
f ∗ g (ω) =
√
2π g (ω)f (ω)
V
V
• Multiplication
If f, g ∈ L1 (R; C) and if ĝ ∈ L1 (R; C), then f g ∈ L1 (R; C) and
1
f g (ω) = √ f ∗ g (ω)
2π
V
V
V
2.8 Summary
51
• Inverse
If f and fˆ ∈ L1 (R; C) , then for almost every t ∈ R
1 Z +iωt ˆ
√
e
f (ω)dω = f (t)
2π R
Or, equivalently
V
V
f (u) = f (−u)
• Transfer formula
If f, g ∈ L1 (R; C), then:
Z
fˆ(t)g(t)dt =
R
Z
f (t)ĝ(t)dt
R
• Plancherel-Parseval
If f, g ∈ L1 (R; C) ∩ L2 (R; C) , fˆ and ĝ ∈ L2 (R; C) and
V
• Some Fourier transforms
T
kfˆk2L2 = kf k2L2
V
(f , g ) = (f, g)
1
ωa
ua (t) −→ √ Sinc
∈ L2 (R; C) ∈
/ L1 (R; C)
2
2π
1
1
∈ L2 (R; C) ∈
/ L1 (R; C)
e−at 1[R+ ] −→ √
a
+
iω
2π
1
2a
e−a|t| ←→ √
∈ L1 (R; C) ∩ L2 (R; C)
2
2π a + ω 2
1
1
2
2
√
e−t /(2a) ←→ √ e−aω /2 ∈ L1 (R; C) ∩ L2 (R; C)
2πa
2π
1
2 ωa
∧a (t) ←→ √ Sinc
∈ L1 (R; C) ∩ L2 (R; C)
2
2π
DR
AF
where a > 0.
T
AF
DR
DR
AF
T
3 — The Schwartz space S(R)
The Schwartz space S(R)
54
The Schwartz space S
It seems quite clear that the Fourier transform in L1 (R; C) has several issues.
Indeed, many of the properties of the Fourier transform summarized in 2.8 are
valid only for a subset of the space of integrable functions L1 . For example, the
properties of differentiation, inversion, multiplication and the Plancherel-Parseval
theorem require not only integrable functions but additional conditions must be
fullfilled for each particular case. This is not very convenient.
AF
T
It turns out that we can approximate every integrable function by functions
that not only are integrable, but that also behave "perfectly" with respect to the
properties of the Fourier transform in L1 (they satisfy all the properties listed in
2.8 without restriction). These functions belong to a functional space called the
Schwartz space S(R; C) ⊂ L1 (R; C). In particular, for every element f ∈ S(R; C),
multiplication by (−it)k (or (iω)k ) for an arbitrary k ∈ N will always correspond
to differentiating k times with respect to ω (or t) in the reciprocal space. In other
words, it is always possible to differentiate under the integral sign for such functions.
In addition, the Schwartz space S has the remarkable property of being invariant
under the Fourier transform, this means that the Fourier transform of any element
of S also belongs to S and therefore the inversion formula always apply in this space.
Now, what are the properties that must be fulfilled by a function f to belong to
the Schwartz space? Of course, for a function f to have derivatives of any order,
f must belong to C ∞ (R; C), so that S(R; C) ⊂ C ∞ (R; C). In addition, it was
already shown in 2.3.4 that the faster an integrable function f decays at infinity,
the smoother its Fourier transform f will be. If we want to make sure that f
has derivatives of any order (f ∈ C ∞ (R; C)), then f must be a rapidly decreasing
function.
DR
3.1
V
V
V
Definition 3.1.1 — Rapidly decreasing function. A function f : mathbbR 7→ C
is said to be rapidly decreasing if it decays to zero at infinity faster than any
inverse power. More precisely, ∀k ∈ N, there exists Ck > 0 such that
∀ t ∈ R,
|t|k |f (t)| ≤ Ck
or, equivalently
sup |t|k |f (t)| < ∞
t∈R
so that tk f is bounded for every k ∈ N.
Definition 3.1.2 — The Schwartz space S(R; C). The Schwartz space S(R; C) is
the space of all functions f : R 7→ C which are infinitely differentiable and such
that f and all of its derivatives are rapidly decreasing.
3.1 The Schwartz space S
55
(
αd
∞
S(R; C) = f ∈ C (R; C) | ∀α, k ∈ N, sup |t
t∈R
k
)
f (t)
|<∞
dtk
(3.1)
Remarks
• In other words, the Schwartz space consists of all functions f such that f
and all of their derivatives exist over R and decay to zero at |t| → ∞ faster
than any inverse power of t.
2
• For a > 0, a gaussian of the form G(t) = e−at , or more generally, any
product of a polynomial and a gaussian belongs to the Schwartz space
2
f (t) = tk e−at ∈ S(R; C)
AF
• If f ∈ S(R; C) and α, k ∈ N, then
T
where k ∈ N.
tα
dk f (t)
∈ S(R; C)
dtk
DR
• If f ∈ S(R; C), then f is p-integrable (f ∈ Lp (R; C)) for any p ∈ [1, ∞] .
Indeed, since f is bounded, f ∈ L∞ (R; C). Moreover, there exists C such
that (1 + t2 )|f (t)| ≤ C, so that for 1 ≤ p < ∞
Z
|f (t)|p dt ≤ C
R
Z
R
dt
< ∞ since 2p > 1
(1 + t2 )p
and f ∈ Lp (R; C). In particular, every element of S(R; C) is integrable
(p = 1) and therefore has a well defined Fourier transform in L1 .
• Every infinitely differentiable function f with compact support (f ∈ Cc∞ (R; C))
belongs to S(R; C). Since Cc∞ (R; C) is dense in Lp (R; C) for 1 ≤ p < ∞, it
follows that the Schwartz space is dense in Lp (R; C).
Definition 3.1.3 — Convergence in S(R; C). Despite the fact that the Schwartz
space is not a normed vector space, it is still possible to define the convergence
in S(R; C) . We say that the sequence (fn )n≥0 of elements of S(R; C) converges
to f ∈ S(R; C) in S(R; C) if
∀α, β ∈ N,
lim kxβ
n→∞
dα
(fn − f )kL∞ = 0
dtα
(3.2)
The Schwartz space S(R)
56
The Fourier transform is a linear bijection S(R; C) 7→ S(R; C)
The Schwartz space has the remarkable property of being invariant under the
Fourier transform, this means F {S(R; C)} = S(R; C). More precisely, the Fourier
transform is a linear bijection of S(R; C) onto itself.
Proof
In order to demonstrate this, let us first show that if f ∈ S(R; C), then its Fourier
transform belongs to the Schwartz space fˆ ∈ S(R; C). For α ∈ N, from (3.5):
V
dα f (ω)
= (−it)α f (ω)
dω α
V
so that, for β ∈ N
V
1
dα f (ω)
ωβ
= β (iω)β (−i)α tα f (ω) = (−1)α (i)α−β (iω)β tα f (ω)
α
dω
i
V
T
and from (3.4)
V
V
AF
dα f (ω)
β
= (−1)α (i)α−β dtd β (tα f )(ω)
ωβ
α
dω
V
β
and since f ∈ S(R; C), dtd β (tα f ) ∈ S(R; C) and its Fourier transform is bounded
(Riemman Lebesgue theorem)
DR
3.2
V
β
sup | dtd β (tα f )| < ∞
ω∈R
and one obtains , for α, β ∈ N
V
dα f (ω)
β
sup |ω β
| = sup | dtd β (tα f )| < ∞
α
dω
ω∈R
ω∈R
V
V
V
which demonstrates that f belongs to the Schwartz space f ∈ S(R; C).
Now, it is clear that the Fourier transform F : S(R; C) → S(R; C) is injective,
since S(R; C) ⊂ L1 (R; C) (see 2.4.1). Finally, since in particular f is integrable,the
inversion theorem (2.8) applies and
V
V
V
f (u) = f (−u)
(3.3)
and one concludes that F : S(R; C) → S(R; C) is surjective, since from (3.3) we
see that every f ∈ S(R; C) is the Fourier transform of an element of S(R; C).
3.3 Properties of the Fourier transform in S(R; C)
3.3
57
Properties of the Fourier transform in S(R; C)
Of course, all the properties of translation, modulation, dilation, contraction,
convolution and the transfer formula, which are valid for every integrable function,
remain valid for any function of the Schwartz space, since S(R; C) ⊂ L1 (R; C). In
addition, all the other properties of the Fourier transform discussed previously
also apply in S(R; C) without any restriction as it will be shown below.
3.3.1
Differentiation
If f ∈ S(R; C), then for any k ∈ N
V
dk f
(ω)
dtk
= (iω)k fˆ(ω)
(3.4)
dk fˆ(ω)
= (−it)k f (ω)
dω k
T
and
(3.5)
AF
V
3.3.2
Inversion
DR
this is a consequence of corollary 2.3.4 and the fact that f and all of its derivatives
are rapidly decaying functions.
This is a consequence of 3.2. If f ∈ S(R; C), its Fourier transform is also in S(R; C)
and the inversion formula always applies:
1 Z +iωt ˆ
√
f (t) =
e
f (ω)dω
2π R
∀t ∈ R
(3.6)
or, equivalently
V
V
f (u) = f (−u)
3.3.3
Multiplication and convolution
If f, g ∈ S(R; C), then
V
g ∗ f (ω) =
√
2π ĝ(ω)fˆ(ω)
which is true since this is valid for every f, g ∈ L1 (R; C). In addition:
(3.7)
The Schwartz space S(R)
58
1
f g (ω) = √ fˆ ∗ ĝ(ω)
2π
V
(3.8)
To prove Eq.(3.8), the bijection of the Fourier transform over S(R; C) assures that
there exists u, v ∈ S(R; C) such that f = u and g = v . Then
V
V
V
V
VV
f g (ω) = uv (ω)
VV
and using Eq.(3.7), uv =
√1 u
2π
V
∗v
1
1
f g (ω) = √ u ∗ v (ω) = √ u ∗ v(−ω)
2π
2π
V
V
V
V
V
and f = u(−ω), g = v(−ω) which proves that
1
f g (ω) = √ fˆ ∗ ĝ(ω)
2π
3.3.4
Plancherel-Parseval
AF
T
V
DR
Since S(R; C) ⊂ L1 (R; C) and S(R; C) ⊂ L2 (R; C), the Plancherel-Parseval theorem (2.7.1) applies in S(R; C). For f, g ∈ S(R; C):
(fˆ, ĝ) = (f, g);
kfˆk2L2 = kf k2L2
(3.9)
We see then that the Schwartz space is the ideal functional space in what concerns
the properties of the Fourier transform in L1 .
3.4
Summary
• The Fourier transform F restricted to the Schwartz space is a linear, continuous and bijective application
F : S(R; C) 7→ S(R; C)
defined as in L1 (since S(R; C) ⊂ L1 (R; C)) by:
1 Z
F {f } (ω) = fˆ(ω) = √
f (t)e−iωt dt
R
2π
• fˆ is of class C ∞ (R), and is a rapidly decreasing function.
3.4 Summary
59
• Translation
V
f (t − a) = e−iaω fˆ(ω)
• Modulation
V
eiat f (t) = fˆ(ω − a)
• Dilation-Contraction
V
V
fa (t) =
1
f (t/a)
|a|
= fˆ(aω)
a ∈ R/{0}
• Differentiation
If f ∈ S(R; C), then for any k ∈ N
V
dk f (t)
(ω)
dtk
= (iω)k fˆ(ω)
• Multiplication by t
If f ∈ S(R; C), then for any k ∈ N
dk fˆ(ω)
dω k
T
V
(−it)k f (ω) =
AF
• Convolution
If f, g ∈ S(R; C), then
V
g ∗ f (ω) =
2π ĝ(ω)fˆ(ω)
DR
• Multiplication
If f, g ∈ S(R; C) then
√
1
f g (ω) = √ f ∗ g (ω)
2π
V
V
V
• Inverse
If f ∈ S(R; C) , then
1 Z +iωt ˆ
e
f (ω)dω
f (t) = √
2π R
Or, equivalently
V
V
f (u) = f (−u)
• Transfer formula
If f, g ∈ S(R; C) , then:
Z
fˆ(t)g(t)dt =
R
Z
f (t)ĝ(t)dt
R
• Plancherel-Parseval
If f, g ∈ S(R; C)
V
V
(f , g ) = (f, g);
kfˆk2L2 = kf k2L2
T
AF
DR
DR
AF
T
4 — Fourier transform in L2(R)
Fourier transform in L2 (R)
62
4.1
Introduction
It is obvious that the definition of the Fourier transform given by Eq.(2.1) can only
be employed for a relatively small set of functions which are integrable. In many
fields, such as quantum mechanics or signal processing, the functional spaces of
square integrable functions L2 play a major role. In this chapter, we will extend
the Fourier transform to the L2 space. Still, restricting the Fourier analysis to
L1 or L2 functions is not enough; functions like the Heavyside "unit step", sin t
or the Dirac δ distribution are not elements of L1 nor L2 . A much more general
extension to the Fourier transform will be discussed in the next chapter.
4.1.1
An intuitive definition
T
Let f be a square-integrable function that is not integrable. In order to define its
Fourier transform, the idea is to approximate f by a sequence of square-integrable
functions (fn )n∈N that are also integrable, fn ∈ L1 (R; C) ∩ L2 (R; C). This is always
possible, for example, one can choose the sequence fn = f 1[n,n] of integrable
functions, since each fn vanish outside a finite interval. This sequence converges
pointwise to f and also under the L2 -norm:
AF
lim kfn − f kL2 = 0
n→∞
V
Since for every n ∈ N, fn is integrable, its Fourier transform fn is well defined :
1 Zn
f (t)e−iωt dt
fn (ω) = √
2π −n
DR
V
and belongs to L2 (R; C) (see the Plancherel-Parseval theorem 2.7.1). It will be
shown in this chapter that the sequence (fn )n∈N converges in L2 to a function
ϕ ∈ L2 (R; C). We define the Fourier transform of f ∈ L2 (R; C) as
F {f } = ϕ
V
∈ L2 (R; C)
so that
1 Zn
lim kF {f } − √
f (t)e−iωt dtkL2 = 0
n→∞
2π −n
4.2
Fourier transform in L2 (R; C)
In order to extend the Fourier transform to the space of square-integrable functions
L2 , we will use the fact that L1 (R; C)∩L2 (R; C) is a dense subset of L2 (R; C). This
means that every square-integrable function of L2 (R; C) is the limit of a sequence
of integrable functions for which the Fourier transform defined in L1 can be applied.
4.2 Fourier transform in L2 (R; C)
63
We recall that a map A : L2 (R; C) → L2 (R; C) is said to be continuous if for any
sequence (fn )n∈N which converges to f in L2 (R; C) (i.e kfn − f kL2 → 0 ), A {fn }
converges to A {f } in L2 (R; C).
Theorem 4.2.1 — Fourier-Plancherel, theorem and definition. There exists a
unique map F : L2 (R; C) → L2 (R; C) which:
a) agrees with the Fourier transform in L1 (R; C) on L1 (R; C) ∩ L2 (R; C)
∀f ∈ L1 (R; C) ∩ L2 (R; C)
V
F {f } = f
b) verifies the Plancherel-Parseval theorem, that is, for every f, g ∈ L2 (R; C)
hf, gi = hF {f } , F {g}i
kf k2L2 = kF {f } k2L2
and
T
c) is continous
(4.1)
DR
F {F {f }} (u) = f (−u)
AF
d) verifies the inversion formula, so that for every f ∈ L2 (R; C) and almost every
u ∈ R:
The map F : L2 (R; C) → L2 (R; C) is, by definition, the Fourier transform in
L2 (R; C).
Remark
From now on, the symbol F represent this "extended" Fourier transform which is
now defined not only in L1 (R; C) but also in L2 (R; C).
Proof
Let f ∈ L2 (R; C). If f ∈ L1 (R; C) ∩ L2 (R; C), we define
V
F {f } = f
where fˆ is the Fourier transform in L1 (R; C). Note that since f ∈ L2 (R; C), F {f }
belongs to L2 (R; C) (see 2.7.1).
Now, if f ∈ L2 (R; C) ∈
/ L1 (R; C), it is always posible to choose a sequence (fn )n∈N
of elements of L1 (R; C) ∩ L2 (R; C) such that
lim kfn − f kL2 = 0
n→∞
Fourier transform in L2 (R)
64
since L1 (R; C) ∩ L2 (R; C) is dense in L2 (R; C). In particular, (fn )n∈N is a Cauchy
sequence in L2 and by applying the Plancherel-Parseval’s theorem (2.7.1)
lim kfj − fk k2L2 = lim kfˆj − fˆk k2L2 = 0
j,k→∞
j,k→∞
so that (fˆn )n∈N is also a Cauchy sequence, and therefore converges in L2 (R; C) (L2
is a complete vector space). In consequence, there exists ϕ ∈ L2 (R; C) such that
lim kfˆn − ϕkL2 = 0
n→∞
we define
F {f } := ϕ
T
By construction, the map F : L2 (R; C) → L2 (R; C) satisfies (a). Now, since
fn → f and fˆn → F {f } in L2 , and given that the inner product is continuous in
L2 , F satisfies the Plancherel-Parseval theorem (b). In particular:
AF
kf k2L2 = kF {f } k2L2
which implies (c), that is, F : L2 (R; C) → L2 (R; C) is a continous map, since
DR
lim kfn − f k2L2 = 0 → lim kF {fn } − F {f } k2L2 = 0
n→∞
n→∞
We will show now that F is unique. Suppose that there exist two continuous
maps F 1 : L2 (R; C) → L2 (R; C) and F 2 : L2 (R; C) → L2 (R; C) that are equal in
L1 (R; C) ∩ L2 (R; C). Then, since (fn )n∈N ∈ L1 (R; C) ∩ L2 (R; C)
V
F 1 {fn } = F 2 {fn } = fn
∀n ∈ N
and taking the limit n → ∞
F 1 {f } = F 2 {f }
∈ L2 (R; C)
since the limit of a sequence in a normed vector space is unique. Finally, since
F {f } ∈ L2 (R; C), the map F can always be applied twice:
F {F {f }} (u) = F F
lim fn
n→∞
(u) = lim F {F {fn }} (u)
n→∞
4.2 Fourier transform in L2 (R; C)
65
where we have used the continuity of F in L2 (R; C) and where the limits are in
the sense fn → f in L2 (R; C). Then, by using the inversion theorem (2.4.1) for
fn ∈ L1 (R; C)
V
V
F {F {f }} (u) = lim fn (u) = lim fn (−u)
n→∞
n→∞
for almost every u ∈ R. Since fn → f in L2 (R; C) , then for almost every u ∈ R
F {F {f }} (u) = f (−u)
Remarks
AF
T
• Whereas the Fourier transform in L1 (R; C) maps L1 (R; C) into C 0 (R; C)
(continuous and bounded functions that decay to zero at infinity), the Fourier
transform in L2 (R; C) maps L2 (R; C) into itself in such a way that the norms
and the inner products are preserved. We see then that the Fourier transform
behaves "better" in L2 (R; C), for example the inversion formula can always
be applied. The definition of the Fourier transform, however, is much less
simple for f ∈ L2 (R; C) than for f ∈ L1 (R; C)
DR
• In a similar way, it is possible to construct F −1 : L2 (R; C) → L2 (R; C), an
extension to L2 of the inverse Fourier transform in L1 defined in 2.8. Then,
for every f ∈ L2 (R; C)
F −1 {F {f }} = f
• We see that F : L2 (R; C) → L2 (R; C) is a bijection (an injection and a
surjection). Indeed, F is an injection since if f, g ∈ L2 (R; C) → L2 (R; C) are
such that F {f } = F {g}, then F {f − g} = 0 and the Plancherel-Parseval
theorem implies that f = g in L2 (R; C). The Fourier transform in L2 is also
surjective, since for every f ∈ L2 (R; C), there exists g ∈ L2 (R; C) such that
F {g} = f . Indeed, the inversion formula states that g is given by
F −1 {f } = g
• Since in addition F preserves the inner product in L2 (R; C), F : L2 (R; C) →
L2 (R; C) is an isomorphism.
• If F {f } is the Fourier transform of f ∈ L2 (R; C), then F {f } is not necessarily continuous over R. The Riemann-Lebesgue theorem is not valid for
the Fourier transform in L2 (R; C).
Fourier transform in L2 (R)
66
Example 4.1 — The Fourier transform of a Sinc. We know that ua ∈ L1 (R; C) ∩
L2 (R; C), so that its Fourier transform in L2 is, by definition, equal to its Fourier
transform in L1 (see example 2.1):
ωa
1
F {ua } (ω) = ua (ω) = √ Sinc
2
2π
V
∈ L2 (R; C) ∈
/ L1 (R; C)
Applying the inversion formula (4.1):
F {F {ua }} (ω) = ua (−ω) = ua (ω)
we find the Fourier transform extended to L2 of a Sinc
(
)
(ω) = ua (ω)
DR
AF
T
F
ta
1
√ Sinc
2
2π
Example 4.2 — The Fourier transform of f (t) = 1/(a + it). The one-sided decaying
exponential g(t) = e−at 1[R+ ] for a > 0 belongs to L1 (R; C) ∩ L2 (R; C). Its Fourier
transform in L2 is equal to its Fourier transform in L1 , which was calculated in
2.3:
n
o
1
1
F e−at 1[R+ ] = e−at 1[R+ ] = √
2π a + iω
V
∈ L2 (R; C) ∈
/ L1 (R; C)
Applying the inversion formula (4.1):
n
n
F F e−aω 1[R+ ]
oo
(ω) = g(−ω) = eaω 1[R− ]
and we find the Fourier transform of 1/(a + it) ∈ L2 (R; C)
(
F
)
1
1
√
(ω) = eaω 1[R− ]
a
+
it
2π
4.3 Properties of the Fourier transform in L2 (R; C)
4.3
67
Properties of the Fourier transform in L2 (R; C)
The translation, modulation, dilation and contraction properties are still valid
for the Fourier transform in L2 (R; C). For example, the translation property
F {f (t − a)} = e−iωa F {f (t)} can be demonstrated as follows.
Let f ∈ L2 (R; C) and (fn )n→∞ ∈ L1 (R; C) ∩ L2 (R; C) a sequence that converges
to f in L2 (R; C). Then
lim kf (t − a) − fn (t − a)kL2 = 0
n→∞
so that
V
lim kF {f (t − a)} − fn (t − a)kL2 = 0
n→∞
V
V
T
but ∀n ∈ N, fn (t − a)(ω) = e−iωa fn (ω), which converges to e−iωa F {f } in L2 (R; C).
This demonstrates the translation property for every f ∈ L2 (R; C):
AF
F {f (t − a)} = e−iωa F {f }
The demonstration of the modulation, dilation and contraction properties can be
done by a similar procedure.
Integral definition of the Fourier transform
DR
4.3.1
If f ∈ L2 (R; C) , its Fourier transform F {f } ∈ L2 (R; C) is such that
1 ZN
lim kF {f } − √
f (t)e−iωt dtkL2 = 0
N →∞
2π −N
Note, however, that in general there is no pointwise convergece:
1 ZN
F {f } (ω) 6= lim √
f (t)e−iωt dt
N →∞
2π −N
The following theorem shows that, in some cases, the poinwise convergence is
verified, which allows to write the Fourier transform in L2 as the limit of an
integral.
Theorem 4.3.1 — Integral definition of the Fourier transform in L2 (R; C) . Let
f ∈ L2 (R; C). If for almost every ω ∈ R the integral
1 ZN
√
f (t)e−iωt dt
2π −N
converges pointwise to a limit ϕ(ω) when N → ∞, then for almost every ω ∈ R
Fourier transform in L2 (R)
68
1 ZN
F {f } (ω) = lim √
f (t)e−iωt dt
N →∞
−N
2π
(4.2)
Proof
Let’s define fN (t) = f (t)1[−N,N ] which belongs to L1 (R; C) ∩ L2 (R; C). Then
1 ZN
F {fN } (ω) = fN (ω) = √
f (t)e−iωt dt
−N
2π
V
∈ L2 (R; C)
The Plancherel-parseval relation (4.2.1) gives
kF {f } − F {fN } k2L2 = kf − fN k2L2 =
Z
|f (t)|2 dt
|t|>N
which tends to zero when N → ∞, then
lim kF {f } − F {fN } k2L2 = 0
T
N →∞
AF
this means that there is a sub-sequence Nj such that Njn→ ∞
o and for which,
for almost every ω ∈ R, there is pointwise convergence F fNj (ω) = fNj (ω) →
V
V
F {f } (ω). But since by hypothesis for almost every ω ∈ R fN (ω) → ϕ(ω) , then
almost everywhere
DR
1 ZN
f (t)e−iωt dt
F {f } (ω) = ϕ(ω) = lim √
N →∞
2π −N
Theorem 4.3.2 — Integral definition of the inverse Fourier transform in L2 (R; C)
. Let f ∈ L2 (R; C). Then F {f } (ω)1[−N,N ] ∈ L1 (R; C) ∩ L2 (R; C) and suppose
its inverse Fourier transform
1 ZN
F {f } (ω)eiωt dω
fN (t) = √
−N
2π
∈ L2 (R; C)
converges for almost every t ∈ R to a limit ϕ(t), then for almost every t ∈ R
1 ZN
√
f (t) = lim
F {f } (ω)eiωt dt
N →∞
2π −N
(4.3)
Proof
If f ∈ L2 (R; C), then F {f } ∈ L2 (R; C) which implies F {f } (ω)1[−N,N ] ∈
L1 (R; C) ∩ L2 (R; C). Then, according to the Plancherel-Parseval theorem
kF {f } − F {f } (ω)1[−N,N ] k2L2 = kf − fN k2L2 =
Z
|ω|≥N
|F {f } (ω)|2 dω
4.4 Fourier transform and differentiation in L2
69
so that
lim kf − fN k2L2 = 0
N →∞
and one can choose a sub-sequence Nj → ∞ such that for almost every t ∈ R,
fNj (t) → f (t) pointwise. But since by hypothesis fN (t) → ϕ(t) almost everywhere,
we have
1 ZN
√
F {f } (ω)eiωt dt
f (t) = ϕ(t) = lim
N →∞
2π −N
Fourier transform and differentiation in L2
T
In this section we will stablish the differentiation properties of the Fourier transform
in L2 . While under certain conditions it is still possible to show that differentiation
with respect to t corresponds to a multiplication by iω in the Fourier domain, it
is in general not true that a multiplication by −it correspond to a differentiation
with respect to ω of the Fourier transform.
Theorem 4.4.1 — Fourier transform and differentiation. Let f ∈ L2 (R; C). If
(
F
AF
f ∈ C 1 (R; C) and f 0 ∈ L2 (R; C), then:
)
df (t)
(ω) = iωF {f } (ω)
dt
(4.4)
DR
4.4
this is analogous to part (b) of theorem 2.3.3 for the Fourier transform in L1 .
Note, however, that there is no equivalent of part (a) of theorem 2.3.3, since
tf ∈ L2 (R; C) does not imply that F {f } is C 1 (R; C).
Proof
Let’s consider a sequence (fn )n→∞ ∈ S(R; C) ⊂ L1 (R; C)∩L2 (R; C) that converges
to f in L2 (R; C) (this is always possible since the Schwartz space is dense in L2 ).
Then:
(
F
dfn
dt
)
V
=
dfn
(ω)
dt
V
= iωfn (ω) = iωF {fn }
∀n ∈ N
so that
(
lim kF
n→∞
df
dt
)
(
−F
)
dfn 2
kL2 = lim kF
n→∞
dt
and one concludes that, in L2 :
(
df
dt
)
− iωF {fn } k2L2 = 0
Fourier transform in L2 (R)
70
(
F
4.5
df
dt
)
= iωF {f }
Fourier transform and convolution
If f ∈ L1 (R; C) and g ∈ L2 (R; C), then the convolution is a square integrable
function, f ∗ g ∈ L2 (R; C) and its Fourier transform is given by
F {f ∗ g} (ω) = F {f } (ω)F {g} (ω)
4.6
(4.5)
Fourier transform and multiplication
If f, g ∈ L2 (R; C), then the product f g is integrable, f g ∈ L1 (R; C) and one has
4.7
T
1
F {f } ∗ F {g} (ω)
2π
Heisenberg’s inequality
(4.6)
AF
V
F {f g} (ω) = f g (ω) =
Theorem 4.7.1 — Heisenberg’s inequality. Let f ∈ L2 (R; C) and f ∈ C 1 (R; C)
Z
2
DR
such that tf, df /dt ∈ L2 (R; C) . Then:
2
1/2
t |f (t)| dt
R
!1/2
df (t) 2
|
| dt
dt
R
Z
≥
1
kf k2L2
2
(4.7)
and this is equivalent to:
Z
1/2 Z
t2 |f (t)|2 dt
R
ω 2 |F{f }(ω)|2 dω
R
1/2
≥ kf k2L2
(4.8)
or, equivalently:
ktf kL2 kωF{f }kL2 ≥ kf k2L2
4.7.1
Proof of the Heisenberg’s unequality
Since f 0 inL2 (R; C), one has (4.4) F = {f 0 } = iωF{f }(ω) and the PlancherelParseval theorem gives
k
df
kL2 = kω 2 F{f }kL2
dt
4.7 Heisenberg’s inequality
71
so that the left term in Eq.(4.7) can be rewritten as in Eq. (4.8). Now, the function
t 7→ t|f (t)|2 is the product of two L2 functions tf and f ∗ and so it is integrable
and decays to 0 an infinity. Its derivative:
d
df ∗
(t|f (t)|2 ) = |f (t)|2 + 2tRe{f
}
dt
dt
is also L1 since tf, df
∈ L2 (R; C). Since t|f |2 and its derivative are integrable, t|f |2
dt
decays to zero at infinity and one has:
Z
R
Z
Z
d
df ∗
2
2
(t|f (t)| )dt = 0 = |f (t)| dt + 2tRe{f
}dt
dt
dt
R
R
and this can be rewritten
0 = kf kL2 + 2Rehf 0 , tf i
T
Using Cauchy-Schwarz inequality (|hx, yi| ≤ kxkL2 kykL2 ):
DR
AF
kf kL2 = −2Rehf 0 , tf i ≤ 2ktf kL2 kf 0 kL2 = 2ktf kL2 kωF{f }kL2
T
AF
DR
DR
AF
T
5 — Fourier transform of tempered dis
Fourier transform of tempered distributions
74
5.1
Introduction
In this chapter we will deal with Fourier analysis of distributions. A distribution
is a generalized function, it can be thought as a "function of functions". More
precisely, a distribution is a linear form that associates a number to every function
of the Schwartz space. We will see that the 1 function, the Heaviside "step" function,
or a sinusoid like sin ωt or cos ωt, can all be assimilated to certain distributions.
The Fourier transform of a distribution which will allows us to generalize the
Fourier analysis to a large class of mathematical objects beyond integrable and
square integrable functions.
5.2
5.2.1
Distributions
Functions with compact support
T
Let f : R → C, the support of f is the closure of the set of values of t for which
f 6= 0, that is
supp(f ) = {t ∈ R | f (t) 6= 0}
AF
A function f : R 7→ C is said to be of compact support if supp(f ) is a compact
(closed and bounded) subset of R 1 . They are therefore zero outside a bounded set.
DR
Remark
Note that in Lp spaces the functions are in reality defined almost everywhere, so
that a proper definition of supp(f ) in such spaces would be
supp(f ) = ∩Ω∈O(f ) (R \ Ω)
with
O(f ) = {Ω open of R | f = 0 almost everywhere ∈ Ω}
The space D(R; C) ⊂ C ∞ (R; C) of infinitely differentiable functions with compact
support plays an essential role in analysis, and in particular in the theory of
distributions.
5.2.2
The space of test functions D(Ω; C)
Let Ω ⊆ R be an open set. A test function over Ω is an infinitely differentiable
function with compact support. The vector space of all test functions over Ω is
denoted D(Ω; C) 2 . A classic example of a test function over R is
1
A function ϕ : Ω 7→ R is said to have compact support in Ω if there exists a compact K ⊂ Ω
such that ϕ(t) = 0 for all t ∈ Ω \ K.
2
also noted Cc∞ (Ω; C) or C0∞ (Ω; C)
5.2 Distributions
75
t 7→

1
 e− 1−t
2

if |t| < 1
0 otherwise
∈ D(R; C)
which is C ∞ and whose support is included in (0, 1).
More generally, if a, b ∈ R and a < b, the function
ϕ(t)
t 7→ ϕ(t) =

1
 e− (x−a)(x−b)

if t ∈]a, b[
0 otherwise
∈ D(R; C)
(5.1)
a
b
is a test function.
Definition 5.2.1 — Convergence in D. In D(Ω; C), a sequence of test functions
T
(ϕn )n∈N converges to ϕ ∈ D(Ω; C) if
AF
• There is a compact set K ⊂ Ω containing the supports of all ϕn , that is
supp(ϕn ) ⊂ K for all n ≥ 1
and we write
DR
• For every α ∈ N, the sequence of derivatives (dα ϕ/dtα )n∈N converges
uniformly to dα ϕ/dtα in Ω when n → ∞.
D
ϕn −
→ϕ
Note finally that D(R; C) is a (dense) subset of the Schwartz space S(R; C).
5.2.3
The space of distributions D0 (Ω; C)
Definition 5.2.2 A distribution over Ω is a linear and continuous form over
D(Ω; C). The vector space of distributions over Ω is therefore the topological
dual of D(Ω), noted D0 (Ω).
In other words, T is a distribution over Ω if the map:
T : D(Ω; C) 7→ C
assigns to each test function ϕ ∈ D(Ω) a complex scalar hT, ϕi such that
• T is linear: for any test functions ϕ1 , ϕ2 and scalars c1 , c2 :
hT, c1 ϕ1 + c2 ϕ2 i = c1 hT, ϕ1 i + c2 hT, ϕ2 i
t
Fourier transform of tempered distributions
76
• T is continuous: for every sequence (ϕn )n∈N ∈ D
D
C
ϕn −
→ ϕ =⇒ hT, ϕn i −
→ hT, ϕi
Equivalently, T is continuous if and only if for every compact K ⊂ Ω there
exists CK > 0 and NK ∈ N such that
∀ϕ ∈ D(Ω; C), supp(ϕ) ⊂ K ⇒ |hT, ϕi| ≤ CK
sup
t∈K,|α|≤NK
Regular distributions
Definition 5.3.1 — Locally integrable functions. A function f is said to be
T
locally integrable over Ω if f is integrable over any compact subset of Ω. The set
of all locally integrable functions over Ω is denoted L1loc (Ω), which includes all
continuous functions and all Lp functions. An important result is the following:
if f ∈ L1loc (Ω) and if ϕ ∈ D(Ω) is a test function, then ϕf is integrable, that is
ϕf ∈ LΩ .
AF
5.3.1
dα ϕ(t)
|
dtα
Examples of distributions
To every locally integrable function f over Ω is possible to associate a distribution
Tf defined by
def
hTf , ϕi =
Z
DR
5.3
|
Ω
f (t)ϕ(t)dt
∈ C ∀ϕ ∈ D(Ω)
(5.2)
Distributions defined in this way from locally integrable functions are called regular distributions. Note that if f and g are two locally integrable functions, then
the associated distributions Tf and Tg are equal if and only if f and g are equal
almost everywhere. If we consider two locally integrable functions which are equal
for almost every value of t to be equivalent, then the linear application f 7→ Tf
from L1loc (Ω) into D0 (Ω) is injective, which allows to write simply Tf = f .
Proof
R
The application ϕ → Ω f (t)ϕ(t)dt is well defined since f is a locally integrable function
and ϕ a test function, and so ϕf is integrable. In order to prove that it defines a
distribution, we must show that the application is linear and continuous. Linearity
is obvious from the properties of the integral. To show that it is continuous, suppos
a sequence (ϕ)n∈N of test functions that converges to ϕ ∈ D(Ω). Then, there is a
compact K ⊂ Ω that contains the support of all the ϕn . The function f being loR
cally integrable, we conclude that M = K |f (t)|dt is a finite number. We then have
R
R
| Ω f (t)(ϕn (t)−ϕ(t))dt| ≤ M Ω |ϕn (t)−ϕ(t)|dt ≤ M V kϕn −ϕk∞ where V is the volume
5.3 Examples of distributions
77
of the compatc K. But kϕn − ϕk∞ tends to zero, and so limn→∞ hf, ϕn − ϕi → 0. By
linearity, we conclude hf, ϕn i → hf, ϕi.
• Heaviside distribution : the Heaviside function H being locally integrable
over R, it defines a regular distribution H : D(R) 7→ C such that, for every
ϕ ∈ D(R):
hH, ϕi =
Z
H(t)ϕ(t)dt =
Z ∞
ϕ(t)dt
0
R
∈C
(5.3)
• The distribution 1 : the constant function t 7→ 1 is locally integrable over
R and it defines the regular distribution 1 : D(R) 7→ C such that, for every
ϕ ∈ D(R):
h1, ϕi =
Z
ϕ(t)dt
∈C
(5.4)
T
R
hsin, ϕi =
Z
sin(t)ϕ(t)dt
∈C
(5.5)
DR
R
5.3.2
AF
• The distribution sin : the function t 7→ sin t defines the regular distribution
sin : D(R) 7→ C such that, for every ϕ ∈ D(R):
The Dirac distribution
Definition 5.3.2 The Dirac distribution δ : D(R) 7→ C is the distribution which
associates to every test function ϕ its value at the origin
def
hδ, ϕi = ϕ(0)
∈ C ∀ϕ ∈ D(Ω)
(5.6)
If a ∈ R, we define the Dirac centered at t = a as the distribution δa : D(R) 7→ C
such that
def
hδa , ϕi = ϕ(a)
∈ C ∀ϕ ∈ D(Ω)
(5.7)
Remark
The Dirac distribution is an example of a non-regular distribution. Sometimes the
notations δ(t) and δ(t − a) are used instead of δ and δa , respectively. However,
it should be remembered that δ(t) does not correspond to the "value" of the
distribution at a point t, since a distribution is not a function. This notation
allows to remember that the independent variable upon which the distribution
Fourier transform of tempered distributions
78
acts is t. One often represents the Dirac distribution as an arrow located at the
point at which it is centered, as shown in the figure below.
δa
δ
t
t
a
The linearity of the Dirac distribution is obvious. To show that it is continuous, suppose a sequence (ϕn )N of test functions that converges to ϕ in
D(R). This means that in particular (ϕn )N converges uniformly to ϕ, and so
limn→∞ ϕn (0) = limn→∞ hδ, ϕn i = ϕ(0) = hδ, ϕi, which proves its continuity.
Definition 5.3.3 The Dirac’s comb comb : D(R) 7→ C is the distribution defined
n=−∞
comb =
∈ C ∀ϕ ∈ D(Ω)
DR
therefore
ϕ(n)
∞
X
(5.8)
AF
∞
X
def
hcomb, ϕi =
T
as
(5.9)
δn
n=−∞
P∞
this distribution is well defined since the sum
because any test function is of compact support.
n=−∞
ϕ(n) is actually finite
comb
t
1
5.3.3
The Cauchy’s principal value
Definition 5.3.4 The function t 7→
1
t
is not locally integrable over R. However,
it is possible to define the distribution p.v 1t : D(R) 7→ C, called "Cauchy’s
principal value of 1/t", by
5.3 Examples of distributions
79
Z
1
ϕ(t)
def
dt
hp.v , ϕi = lim+
→0
t
|t|>
t
5.3.4
∈ C ∀ϕ ∈ D(Ω)
(5.10)
Restriction of a distribution
Definition 5.3.5 Let ω ⊂ Ω be two open sets and T ∈ D 0 (Ω) a distribution over
∈ D0 (ω) defined by:
Ω. The restriction of T to ω is the distribution T
ω
def
hT , ϕi = hT , ϕ̃i
∀ϕ ∈ D0 (Ω)
ω
where
ϕ̃(t) =
ϕ(t) if t ∈ ω
0 otherwise
(5.11)
T
(
AF
For example, let Ω =]0, ∞[. Then
H
=1
Ω
Support of a distribution
DR
5.3.5
Ω
Let Ω be an open subset of R. We say that the distribution T is zero in U ⊂ Ω if,
for every ϕ ∈ D(Ω) whose support is in U , we have
hT, ϕi = 0
Definition 5.3.6 The support supp(T) of a distribution T over Ω is the com-
plementary of the largest open set on which T is zero. Equivalently, it is the
smallest closed set outside which the distribution is zero:
supp(T) =
\
E
n
o
with F (T) = E closed ∈ Ω , T|Ω\E = 0
(5.12)
E∈F (T)
We note E 0 (Ω) ⊂ D0 (Ω) the sub-space of distributions with compact support in
Ω.
For example, for every a ∈ R, supp(δa ) = {a} and supp(comb) = Z.
Fourier transform of tempered distributions
80
5.3.6
Convergence in D0 (Ω)
Definition 5.3.7 We say that the sequence of distributions (Tn )n≥0 converges to
T in D0 (Ω) if
∀ϕ ∈ D(R),
lim |hTn , ϕi − hT, ϕi| = 0
n→∞
Example 5.1 — Approximation of the Dirac distribution. In example 2.2 we
discussed the sequence of square functions (fn = Π1/n )n∈N . As n increases, fn
gets more and more concentrated near zero. Of course, this sequence converges
pointwise to 0 for every t =
6 0, but it was shown that when Π1/n is inside an
integral, it concentrates the weight of the integral at the origin as n → ∞, in the
sense:
lim
n→∞
Z
Π1/n (t)h(t) = h(0)
R
lim hΠ1/n , ϕi = lim
n→∞
n→∞ R
Π1/n (t)ϕ(t)dt = ϕ(0) = hδ, ϕi
DR
and we write
Z
AF
T
where h is a continuous function. If we consider, for each u1/n the associated regular
distribution Π1/n , we see from Eq.(5.6) that the sequence (Π1/n )n∈N converges to
the Dirac distribution. Indeed
∈ C ∀ϕ ∈ D(R)
D0
Π1/n −→ δ
5.4
Operations on distributions
In the following, we would like to extend the operations defined for functions
(translation, dilation/contraction, change of variables, differentiation and, of course,
the Fourier transform) to distributions. This is made in the following way
• Let L : L1loc (Ω) 7→ L1loc (Ω) be a transformation over the space of locally
integrable functions.
• For every locally integrable function f ∈ L1loc (Ω) we look for a transformation
L0 over the space of test functions such that
hL(f ), ϕi = hf , L0 ϕ) ∀ϕ ∈ D(Ω)
where L(f ) is the regular distribution defined by the locally integrable function L(f ).
5.4 Operations on distributions
81
• We define, for every distribution T, its transformation L(T) by the formula
hL(T), ϕi = hT, L0 ϕ) ∀ϕ ∈ D(Ω)
Translation of a distribution
Let f ∈ L1loc (R) be a locally integrable function and a ∈ R. The translation τa f
of the function f is
def
τa f (t) = f (t − a)
the regular distribution associated to this function, noted τa f verifies:
hτa f , ϕi =
Z
f (t − a)ϕ(t)dt =
R
Z
f (t)ϕ(t + a)dt = hf , τ−a ϕi
R
and this can be generalized to any distribution, as follows.
T
Definition 5.4.1 The distribution T translated by a ∈ R, and noted τa (T) is
def
hτa (T ), ϕi = hT , τ−a ϕi
Examples
AF
defined by
∀ϕ ∈ D(R)
(5.13)
DR
5.4.1
• The Dirac distribution τa (δ) translated by a ∈ R is given by
hτa (δ), ϕi = hδ, τ−a ϕi = ϕ(a) = hδa , ϕi ∀ϕ ∈ D(R)
and so, intuitively, the Dirac translated by a is the Dirac centered at t = a
τa (δ) = δa
1
• The Cauchy’s principal value translated by a ∈ R, noted τa (p.v 1t ) = p.v t−a
is
hp.v
1
1
, ϕi = hp.v , τ−a ϕi = lim+
→0
t−a
t
Z
|t|>
ϕ(t + a)
dt ∀ϕ ∈ D(R)
t
and after a change of variables:
hp.v
1
t−a
, ϕi = lim+
→0
Z a−
−∞
!
Z ∞
ϕ(t)
ϕ(t)
dt +
dt
t−a
a+ t − a
∀ϕ ∈ D(R)
Fourier transform of tempered distributions
82
5.4.2
Dilation/Contraction of a distribution
Let f ∈ L1loc (R) be a locally integrable function and a ∈ R \ 0. The function f
dilated by a, denoted ∂a f is
def
∂a f (t) =
1
t
f( )
|a| a
the regular distribution associated to this function, noted ∂a f verifies:
h∂a f , ϕi =
Z
R
1
t
1 Z
f ( )ϕ(t)dt =
f (t)|a|ϕ(at)dt
|a| a
|a| R
1 Z
1
h∂a f , ϕi =
hf , ∂1/a ϕi
f (t)(∂1/a ϕ)(t)dt =
|a| R
|a|
∀ϕ ∈ D(R)
∀ϕ ∈ D(R)
and this can be generalized to any distribution, as follows.
1
∂1/a ϕi
|a|
AF
h∂a (T ), ϕi = hT ,
T
Definition 5.4.2 The dilation by a of a tempered distribution T is defined by:
∀ϕ ∈ D(R)
(5.14)
Example 5.2 — Dilation of the Dirac’s comb. The Dirac’s comb ∂a (comb) di-
DR
lated by a ∈ R \ {0} is defined as
h∂a (comb), ϕi = hcomb,
∞
∞
X
X
1
∂1/a ϕi =
ϕ(na) =
hδna , ϕi
|a|
n=−∞
n=−∞
which is a Dirac’s comb of period a
∞
X
∂a (comb) =
δna
n=−∞
5.4.3
Derivative of a distribution
Let f ∈ L1loc (R) be a differentiable, locally integrable function, such that f 0 is
locally integrable as well. Then, the regular distribution f 0 associated to f 0 is such
that
hf 0 , ϕi =
Z
R
f 0 (t)ϕ(t)dt = [f (t)ϕ(t)]∞
−∞ −
|
{z
=0
}
Z
R
f (t)ϕ0 (t)dt = −hf , ϕ0 i
∀ϕ ∈ D(R)
5.4 Operations on distributions
83
where f (t)ϕ(t) vanishes at t → ±∞ since ϕ is of compact support. If f is m times
differentiable with locally integrable derivatives, then one obtains
Z
Z
dm f
dm f (t)
dm ϕ
dm ϕ(t)
m
m
h m , ϕi =
ϕ(t)dt
=
(−1)
dt
=
(−1)
hf
,
i
f
(t)
dt
dtm
dtm
R dtm
R
∀ϕ ∈ D(R)
and this can be generalized to any distribution, as follows.
Definition 5.4.3 The derivative of a distribution T ∈ D 0 (R) is the distribution
D1 T ∈ D0 (R) defined by:
hD1 T , ϕi = −hT , ϕ0 i
def
∀ϕ ∈ D(R)
(5.15)
dm
ϕi
dtm
AF
def
hDm T , ϕi = hT , (−1)m
T
Moreover, since any test function has a derivative of order m ∈ N, which is
itself another test function, it is always possible to define Dm T , the derivative
of order m ∈ N of a distribution T as:
∀ϕ ∈ D(R)
(5.16)
DR
A distribution is therefore infinitely differentiable.
Example 5.3 — Derivative of the Dirac distribution. From (5.15), the derivative
of the δ distribution satisfies:
hD1 δ, ϕi = hδ, −ϕ0 i = −ϕ0 (0)
∀ϕ ∈ S(R; C)
which associates to any ϕ ∈ D(R) the value -ϕ0 (0) ∈ C.
Example 5.4 — Dipole. An electric dipole consists of two point charges of
opposite sign located very close to each other, and one usually takes the limit
when this distance goes to zero by keeping a finite dipole moment µ at the origin.
Consider therefore the distribution over R
Th = µ
1
δh/2 − δ−h/2
h
It is easy to show that, when h → 0
D0
Th −→ −µD1 δ
Indeed
Fourier transform of tempered distributions
84
hTh , ϕi = µ hδh/2 , ϕi − hδ−h/2 , ϕi =
µ
(ϕ(h/2) − ϕ(h/2))
n
so that
µ
(ϕ(h/2) − ϕ(h/2)) = ϕ0 (0) = −µhD1 δ, ϕi
h→0 n
lim hTh , ϕi = lim
h→0
Example 5.5 — Derivative of the Heaviside distribution. Let us consider the
Heaviside H distribution. According to (5.15), its derivative D1 H is given by:
hD1 H, ϕi = hH, −ϕ0 i = −
Z ∞
0
dϕ(t)
dt = ϕ(0)
dt
∀ϕ ∈ D(R)
We see then that the derivative of the Heaviside distribution is a Dirac distribution:
(5.17)
AF
D1 H = δ
∀ϕ ∈ D(R)
T
hD1 H, ϕi = ϕ(0) = hδ, ϕi
Example 5.6 — Derivative of the Sign distribution. Let us consider the sign
DR
distribution. According to (5.15):
hD1 sign, ϕi = hsign, −ϕ0 i = −
Z ∞
0
We see that:
Z 0
dϕ(t)
dϕ(t)
dt +
dt = 2ϕ(0)
dt
dt
−∞
∀ϕ ∈ D(R)
D1 sign = 2δ
This could have been obtained directly by noticing that sign = 2H − 1 and that
D1 1 = 0.
Example 5.7 — Derivative of the rect distribution. The derivative of the Π
distribution verifies:
hD1 Π, ϕi = hΠ, −ϕ0 i =
Z 1/2
−1/2
dϕ(t)
dt = ϕ(−1/2) − ϕ(1/2)
dt
∀ϕ ∈ D(R)
so that
D1 Π = δ−1/2 − δ1/2
5.4 Operations on distributions
85
In examples 5.5, 5.6 and 5.7 we see how the distributions allows us to generalize
the concept of derivative. When looked at as functions, the Heaviside, sign and
rect functions are not differentiable at their discontinuities. In the framework
of distributions, we see that their derivatives are well defined and correspond to
dirac distributions centered at the discontinuities of the respective functions, and
multiplied by the size of the jumps. This can be generalized for functions having
an arbitrary number of jumps, as shown below.
Theorem 5.4.1 — Derivative of a discontinuous function and the jump formula.
Suppose a function f which is C 1 (R) piecewise, having only discontinuities of
the first kind (also called jump discontinuities) a that we note
a1 < a2 < ... < an
the derivative of the distribution f is given by the jump formula
n
X
(f (ak + 0) − f (ak − 0)) δak
(5.18)
T
D1 f = f 0 +
k=1
AF
where f 0 is the distribution defined by the locally integrable function t 7→ f 0 (t).
a
DR
f has
S a discontinuity of first kind at a if there is > 0 such that f is continuous over
]a − , a[ ]a, a + [, f admits finite limits at the left and at the right of a, noted f (a − 0) and
f (a + 0), respectively, and f (a − 0) 6= f (a + 0).
Proof
Let ϕ ∈ S(R; C) and calculate
−
Z
f (t)ϕ0 (t)dt = −
Z a1
f (t)ϕ0 (x)dx −
−∞
R
n−1
X Z ak+1
f (t)ϕ0 (x)dx −
k=1 ak
Z ∞
f (t)ϕ0 (t)dt
an
integration by parts yields
−
Z ak+1
ak
a
f (t)ϕ0 (t)dt = −[f (t)ϕ(t)]akk+1 +
Z ak+1
f 0 (t)ϕ(t)dt
ak
= − (f (ak+1 − 0)ϕ(ak+1 ) − f (ak + 0)ϕ(ak )) +
Z ak+1
ak
and similarly
−
Z a1
−∞
f (t)ϕ (t)dt = −f (a1 − 0)ϕ(a1 ) +
Z ∞
=
an
0
f (t)ϕ0 (t)dt = f (an + 0)ϕ(an ) +
Z a1
f 0 (t)ϕ(t)dt
−∞
Z ∞
an
f 0 (t)ϕ(t)dt
f 0 (t)ϕ(t)dt
Fourier transform of tempered distributions
86
so that
−
Z
f (t)ϕ0 (t)dt = −f (a1 − 0)ϕ(a1 ) + f (an + 0)ϕ(an )
R
+
n−1
X
(f (ak + 0)ϕ(ak ) − f (ak+1 − 0)ϕ(ak+1 ))
k=1
Z a1
+
f 0 (t)ϕ(t)dt +
−∞
n−1
X Z ak+1
f 0 (t)ϕ(t)dt +
Z ∞
an
k=1 ak
|
R
R
f 0 (t)ϕ(t)dt
{z
}
f 0 (t)ϕ(t)dt
We find
−
Z
f (t)ϕ0 (t)dt =
R
|
{z
}
hD1 f ,ϕi
Z
f 0 (t)ϕ(t)dt +
|R
{z
}
hTf 0 ,ϕi
n
X
(f (ak + 0) − f (ak − 0)) ϕ(ak )
| {z }
k=1
hδak ,ϕi
T
Example 5.8 — Derivative of ln |t|. The function t 7→ ln |t| is locally integrable
and therefore it defines the tempered distribution ln |t|. One has, for every
ϕ ∈ S 0 (R; C):
AF
hD1 ln |t|, ϕi = −hln |t|, ϕ0 i = −
Z 0
DR
hD1 ln |t|, ϕi = −
Z
ln |t|ϕ0 (t)dt −
ln |t|ϕ0 (t)dt
R
Z ∞
−∞
and
−
Z ∞
0
ln |t|ϕ (t)dt = lim
Z ∞
→0 0
−
Z ∞
(
∞
0
ln tϕ (t)dt = lim [− ln tϕ(t)]
→0
0
ln |t|ϕ (t)dt = lim ln ϕ() +
+
Z ∞
Z ∞
→0
0
ln |t|ϕ0 (t)dt
0
)
ϕ(t)
dt
t
!
ϕ(t)
dt
t
but ln (ϕ() − ϕ(0)) tends to 0 as 0 → 0 since |ϕ() − ϕ(0)| ≤ max|ϕ0 | and
ln → 0. Then
−
Z ∞
0
ln |t|ϕ (t)dt = lim ln ϕ(0) +
Z ∞
→0
0
!
ϕ(t)
dt
t
in the same way one obtains
−
Z 0
−∞
ln |t|ϕ0 (t)dt = lim − ln ϕ(0) +
→0
Z −
−∞
!
ϕ(t)
dt
t
5.4 Operations on distributions
87
Finally
Z ∞
−
0
ln |t|ϕ (t)dt = lim
−∞
Z
→0 |t|≥
(Z
)
ϕ(t)
ϕ(t)
dt = p.v
dt
t
t
R
and
D1 ln |t| = p.v
1
t
Multiplication by a C ∞ function
hχf , ϕi =
Z
AF
T
It is not possible to define in general the multiplication between two distributions.
This is true even for regular distributions: if f and g are two locally integrable
functions, their product f g is not necessarilly locally integrable. However, if χ
is a function of class C ∞ (R), then for every test function ϕ ∈ D(R) the product
χϕ ∈ D(R) is a test function as well, and in particular locally integrable. Let f
f ϕ ∈ L1loc (R) be a locally integrable function, and χf the regular distribution
associated to χf , then
(χ(t)f (t))ϕ(t)dt =
R
Z
f (t)(χ(t)ϕ(t))dt = hf , χϕi
∀ϕ ∈ D(R)
R
and this can be generalized to any distribution, as follows.
DR
5.4.4
Definition 5.4.4 The multiplication of a distribution T by an infinitely differen-
tiable function χ ∈ C ∞ (R) is the distribution χT defined by
def
hχT , ϕi = hT , χϕi
∀ϕ ∈ D(R)
(5.19)
Example 5.9 — Dirac multiplied by a C ∞ function. Let us consider a function
f ∈ C ∞ (R and δa the Dirac distribution centered at a. The distribution f δa is
such that:
hf δa , ϕi = hδa , f ϕi = f (a)ϕ(a) = f (a)hδa , ϕi
and we obtain, by linearity, the following result:
f δa = f (a)δa
One concludes, in addition, that f comb is given by:
f comb =
∞
X
n=−∞
f (n)δn
∀ϕ ∈ D(R)
Fourier transform of tempered distributions
88
In particular, for χ = t we have:
tδ = 0
A generalization of this latter result will be given later.
Example 5.10 — Multiplication of t and v.p 1t . Let us consider the distribution
tp.v 1t consisting on the multiplication of t ∈ C ∞ (R) by the principal value of 1/t,
p.v 1t :
1
1
Z
ϕ(t)
htp.v , ϕi = hp.v , tϕi = p.v t
dt = ϕ(t)dt = h1, ϕi
t
R
R
t
t
Z
∀ϕ ∈ D(R)
1
t
=1
AF
tp.v
T
And we get the following remarquable result: the "inverse" of t is the principal
value of 1/t:
Recall that 1/t does not define a regular distribution.
DR
Remark The continuity of the function χ is essential to define χT where T is a
distribution. For example, is is not possible to define Hδ or HH = H 2 . If it was
the case, one would have
δ = D1 H = D1 (H 2 ) = 2Hδ = D1 (H 3 ) = 3H 2 δ = 3Hδ
and one could conclude that 2 = 3.
Theorem 5.4.2 In the framework of distributions, the solutions to the equation
tT = 1 are the family of distributions of the form
1
T = p.v + αδ
t
α∈C
(5.20)
Proof
Let S = T − p.v 1t . Then, tS = 0 and we conclude that S is a multiple of δ.
Theorem 5.4.3 Let n ∈ N+ . The solutions of the equation tn T = 0 are the linear
combinations of {δ, δ 0 , ..., δ n−1 }
Every distribution with compact support is a tempered distribution.
5.5 The space of tempered distributions S 0 (R)
89
Theorem 5.4.4 Let Ω be an open subset of R, T ∈ D(R) and x0 ∈ Ω. If
supp(T) ⊂ {t0 }, then T is a finite linear combination of δt0 and its derivatives.
In other words, there exists (λα )α∈N such that λα = 0 except for a finite number
of α’s and
T=
X
λα Dα δt0
α∈N
From now on, we will focus on a subset of D0 (R), the set of tempered distributions,
by extending continuously to a larger space of test functions, the Schwartz space
S(R; C). A main reason to focus on such distributions is that every tempered
distribution has a Fourier transform (to be defined later) which is, in turn, a
tempered distribution. This is due to the fact that the Fourier transform F maps
S into itself.In contrast, the Fourier transform ϕ̂ of a test function ϕ ∈ D(R; C)
cannot have compact support so ϕ̂ does not belong to D(R; C). In consequence,
not every element of D0 (R) has a Fourier transform.
T
The space of tempered distributions S 0 (R)
Definition 5.5.1 The space of tempered distributions S 0 (R) is defined as the
AF
space of all linear, continuous forms that map S(R; C) into C. In other words,
T is a tempered distribution if the map:
T : S(R; C) 7→ C
is linear and
DR
5.5
S
C
ϕn −
→ ϕ =⇒ T(ϕn ) −
→ T(ϕ)
Since D(R; C) ⊂ S(R; C), every tempered distribution T ∈ S 0 (R) defines by
restriction a distribution in D0 (R; C). Therefore
S 0 (R) ⊂ D0 (R)
Examples of tempered distributions
• The dirac distribution (5.7), the heaviside (5.3) and the principal value
distribution (5.10) are all tempered distributions.
• There are different class of functions that define tempered distributions by
means of an integral.
– If u ∈ L1loc (R; C), then the linear form u : S(R; C) → (R) defined as:
hu, ϕi =
Z
u(t)ϕ(t)dt
R
is a tempered distribution.
∈ C ∀ϕ ∈ S(R; C)
(5.21)
Fourier transform of tempered distributions
90
– Any p-integrable function u ∈ Lp (R; C) for p ≥ 1 defines a tempered
distribution since Lp (R; C) ⊂ L1loc (R; C).
– Let u : R → C be a "slowly increasing" function, that is there exists
ku ∈ N such that u/(1 + t2 )ku ∈ L1 (R; C). Then u defines a tempered
distribution. Indeed, uϕ can be rewritten as a product of (1 + t2 )−ku u
which is integrable by definition and (1 + t2 )ku ϕ which belongs to L∞
since ϕ ∈ S
• The distribution over R
T=
X
ak δk
k∈Z
is tempered if the sequence (ak )k∈Z is of polynomial growth, that is if there
is an integer p ≥ 0 such that
ak = O(|k|p ) when |k| → ∞
Definition 5.5.2 — Convergence in S 0 . We say that the sequence of tempered
distributions (Tn )n≥0 converges to T in S 0 if
T
Linear applications of tempered distributions
How to define the derivative or the Fourier transform of a tempered distribution?.
How to translate and dilate a distribution?. In order to define all these operations,
we will note that it is possible to define linear applications over S 0 from linear
applications that map S(R; C) into S(R; C).
DR
5.6
lim |hTn , ϕi − hT, ϕi| = 0
n→∞
AF
∀ϕ ∈ S(R; C),
Definition 5.6.1 — Transpose of a linear map. Let L : S(R; C) 7→ S(R; C) be
a linear continuous map (for example, L can be the Fourier transform in L1
acting on the Schwartz space). The transpose map, denoted by Lt , is defined
over S 0 by
hLt T , ϕi = hT , Lϕi
(5.22)
it can be shown that Lt T is a tempered distribution, so that Lt maps S 0 into
S 0.
In order to show that Lt : S 0 → S 0 , let us suppose that (ϕn )n≥0 is a sequence of
S
elements of S(R; C) such that ϕn −
→ ϕ. Since L is a continuous map, we have
S
L(ϕn ) −
→ L(ϕ). Let T ∈ S 0 be a tempered distribution, then
C
hT , L(ϕn )i −
→ hT , Lϕi
S
So that ϕn −
→ ϕ implies
5.6 Linear applications of tempered distributions
91
hLt T , ϕn i = hT , L(ϕn )i −
→ hT , L(ϕ)i = hLt T , ϕi
C
which proves that Lt T : S(R; C) → C is a linear continuous map, and therefore
Lt T is a tempered distribution.
We see that if L = F represents the Fourier transform in S(R; C), its transposed
application F t : S 0 → S 0 will be, by definition, the Fourier transform over the
space of tempered distributions. First, let us show that any transposed application
is a linear, continuous application.
Proposition 5.6.1 A transpose map Lt : S 0 → S 0 is linear and continuous.
T
Proof
The linearity of Lt is a direct consequence of the fact that L is a linear map. In
S0
order to show that Lt : S 0 → S 0 is continuous, let’s suppose that (Tn )n≥0 −
→ T.
Then, for every ϕ ∈ S(R; C)
lim hLt Tn , ϕi = n→∞
lim hTn , L(ϕ)i = hT , L(ϕ)i = hLt T , ϕi
n→∞
AF
S0
which shows that Lt Tn −
→= Lt T and Lt : S 0 → S 0 is therefore a continuous
application.
DR
Proposition 5.6.2 — Linear, continuous applications over the Scwhartz space
S(R; C) . The following are linear continuous applications L : S(R; C) → S(R; C)
• Translation ϕ 7→ τa ϕ(t) := ϕ(t − a) a ∈ R
• Dilation/Contraction ϕ 7→ ∂λ ϕ(t) :=
• Differentiation ϕ 7→
dk ϕ(t)
dtk
1
ϕ(t/λ)
|λ|
λ ∈ R \ {0}
k∈N
• Multiplication ϕ 7→ Xϕ
where X belongs to the space of tempered functions OM (R; C)
(
∞
OM (R; C) = X ∈ C (R; C), ∀α ∈ N, ∃kα ∈ N such that (1 +
• Convolution ϕ 7→ f ∗ ϕ f ∈ S(R; C)
• Fourier transform ϕ 7→ ϕ
V
5.6.1
Multiplication of a distribution by a tempered function
5.6.2
Convolution
One can verify that for every u, v, ϕ ∈ S(R; C) one has:
dα X
t)−kα α
dt
)
∞
∈ L (R; C)
Fourier transform of tempered distributions
92
Z
(ϕ ∗ u) (t)v(t)dt =
Z Z
R
R
ϕ(t − s)u(s)ds v(t)dt
R
and by Fubini’s theorem
Z
(ϕ ∗ u) v(t)dt =
R
Z Z
R
ϕ(t − s)v(t)dt u(s)ds =
Z
R
(ϕ ∗ v) (−s)u(s)ds
R
this result can be generalized to define the convolution between a distributions
and a tempered function.
Definition 5.6.2 — Convolution of a distribution by a tempered function. For
every tempered distribution T and φ ∈ S(R; C), one defines the convolution
T ∗ φ ∈ S 0 (R; C) by:
hT ∗ φ, ϕi = hT , ∂−1 φ ∗ ϕi
(5.23)
Example 5.11 — Convolution by a Dirac. Let f ∈ S(R; C), according to (5.23),
T
∀ϕ ∈ S(R; C)
hδ∗f, ϕi = hδ, ∂−1 f ∗ϕi =
Z
R
AF
the convolution δ ∗ f is a tempered distribution given by:
f (τ )ϕ(τ − t)dτ
=
Z
t=0
f (τ )ϕ(τ )dτ = hf , ϕi
∀ϕ ∈ S(R; C)
R
DR
where f is the tempered distribution associated with f . We see that the δ is a
neutral element for the convolution product:
δ∗f =f
Definition 5.6.3 — Convolution of a tempered distribution by distribution of
compact support. For every distribution of compact support S ∈ E 0 (R) and
every tempered distribution T ∈ S 0 (R) the convolution product is the tempered
distribution T ∗ S ∈ S 0 (R) defined by
hT ∗ S, ϕi = hT, (∂−1 S) ∗ ϕi = hS, (∂−1 T) ∗ ϕi
∀ϕ ∈ S(R, C)
(5.24)
where T ∗ ϕ(x) = hT, ϕ(x − ·)i = hT, τ−x ∂−1 ϕi
5.6.3
Differentiation and Convolution
For every T ∈ S 0 (R; C) and f ∈ S(R; C):
Dα (T ∗ f ) = (Dα T ) ∗ f = T ∗ (
dα
f)
dtα
(5.25)
5.7 The Fourier transform of tempered distributions
5.6.4
93
Solution of differential equations with distributions
Let us consider a linear system of "input" f and "output" u in which both are
related by a differential equation
Lu = f
(5.26)
in which the differential operator L is of the form
L=
X
i∈I
Ci
dαi
dtαi
where αi ∈ N, I is of finite cardinality and Ci ∈ C. Equation (5.26) can be
extended to the more general framework of tempered distributions:
Lu = f
L=
X
C i D αi
(5.27)
T
i∈I
AF
The tempered distribution h ∈ S 0 (R, C) is a fundamental solution of the differential operateur L if
∈ S 0 (R, C)
DR
L(h) = δ
Corollary 5.6.3 For every f ∈ S(R, C), the solution of the differential equation
Lu = f is given by the convolution:
u=f ∗h
Indeed, one has:
L(u) =
X
Ci Dαi (f ∗ h) = f ∗
i∈I
X
Ci Dαi h = f ∗ L(h)
i∈I
and since L(h) = δ:
L(u) = f ∗ δ = f
5.7
The Fourier transform of tempered distributions
Let us recall the transfer formula (2.13 ) applied to the Schwartz space:
Z
R
fˆ(t)ϕ(t)dt =
Z
R
f (t)ϕ̂(t)dt
∀f, ϕ ∈ S(R; C)
Fourier transform of tempered distributions
94
Since fˆ belongs to S(R; C), it defines a tempered distribution fˆ and from the
transfer formula it is seen that:
hfˆ, ϕi = hf , ϕ̂i
∀ϕ ∈ S(R; C)
It seems natural to define the distribution fˆ as being the Fourier transform of
the distribution f . This result can be generalized in order to define the Fourier
transform of any tempered distribution.
Definition 5.7.1 — Fourier transform in S 0 (R; C). The Fourier transform F :
S 0 (R; C) 7→ S 0 (R; C) in the space of tempered distributions is defined as the
transpose 5.22 of the Fourier transform acting on the Scharwz space .The Fourier
transform F {T } of the distribution T is given by:
5.8
∀ϕ ∈ S(R; C)
(5.28)
T
hF {T } , ϕi = hT , ϕ̂i
The inverse Fourier transform
AF
Theorem 5.8.1 — Inverse Fourier transform. The Fourier transform F : S 0 (R; C) 7→
S 0 (R; C) is a linear, continuous and bijective application of S 0 (R; C) into itsel.
In addition, one has the inversion formula
5.8.1
Proof
F {F {T }} = ∂−1 T
DR
∀ T ∈ S 0 (R; C),
(5.29)
The Fourier transform is a linear and continuous application since is the transpose
of a linear continuous map (the Fourier transform in the Scharwz space). To
demonstrate the inversion formula, one has ∀ϕ ∈ S(R; C):
V
V
hF {F {T }} , ϕi = hF {T } , ϕ̂i = hT , f (ϕ)i
V
V
and by using the inversion formula in S, f (ϕ) = ∂−1 ϕ so that:
hF {F {T }} , ϕi = hT , ∂−1 ϕi = h∂−1 T , ϕi
Finally, the bijection F : S 0 (R; C) 7→ S 0 (R; C) follows from the inversion formula
5.29.
Example 5.12 — Fourier transform of a Dirac. The definition of the Fourier
transform in S 0 allows us now to calculate the Fourier transform of a Dirac
distribution. For every ϕ ∈ S(R; C)
5.8 The inverse Fourier transform
95
hF {δ} , ϕi = hδ, ϕ̂i = ϕ̂(0)
and since ϕ is integrable:
1 Z
1
hF {δ} , ϕi = ϕ̂(0) = √
ϕ(t)dt = √ h1, ϕi
2π R
2π
So√the Fourier transform of δ is the distribution obtained from a constant function
1/ 2π:
1
F {δ} = √ 1
2π
Conversely, applying the inversion formula 5.29:
(
F {F {δ}} = F
)
1
√ 1 = ∂−1 δ = δ
2π
AF
T
1
δ ←→ √ 1
2π
DR
Remark
The constant function equal to 1 is not integrable and therefore as a function it
does not have a Fourier transform. However, it does define a tempered function 1
and in this framework, it has a well-defined Fourier transform:
F {1} =
√
2πδ
This illustrates the interest of extending the Fourier transform to the space of
tempered distributions.
Example 5.13 — Fourier transform of Sin. The Fourier transform of the distribu-
tion sin is such that:
hF {sin} , ϕi = hsin, ϕ̂i =
Z
sin(ω)ϕ̂(ω)dω
ϕ ∈ S(R; C)
R
and using the Euler identity sin(ω) = 1/2i(eiω − e−iω )
√
2π
1 Z iω
−iω
hF {sin} , ϕi =
e −e
ϕ̂(ω)dω =
(ϕ(1) − ϕ(−1))
2i R
2i
where we have used the inversion formula for ϕ. Finally
√
hF {sin} , ϕi =
2π
(hδ1 , ϕi − hδ−1 , ϕi)
2i
ϕ ∈ S(R; C)
Fourier transform of tempered distributions
96
and we get the identity
√
F {sin} =
2π
(δ1 − δ−1 )
2i
5.9
Compatibility with the Fourier transform in L1 and L2
If f ∈ L1 (R, C), its Fourier transform in L1 is a continous and bounded function,
F {f } = fˆ ∈ Cb (R, C). If f ∈ L2 (R, C) , its Fourier transform in L2 is squareintegrable, F {f } ∈ L2 (R, C). In both cases, the Fourier transform of f defines
a tempered distribution F {f }. It can be shown that it is equal to the Fourier
transform of the distribution f as defined in 5.28:
Indeed:
hF {f } , ϕi =
Z
T
F {f } = F {f }
F {f } (ω)ϕ(ω)dω ∀ϕ ∈ S(R; C)
AF
R
and using the transfer formula (2.13):
hF {f } , ϕi =
Z
f (t)ϕ̂(t)dt = hf , ϕ̂i = hF {f } , ϕi ∀ϕ ∈ S(R; C)
5.10
DR
R
Properties of the Fourier transform in S 0
Theorem 5.10.1 — Fourier transform and translation. Let T ∈ S 0 (R; C). Then,
for any a ∈ R
F {τa T } = e−iωa F {T }
a∈R
(5.30)
Proof
Indeed, one has for every ϕ ∈ S(R; C)
hF {τa T } , ϕi = hτa T , ϕ̂i = hT , τa ϕ̂i
and from the translation properties of the Fourier transform inthe Schwartz space,
τa ϕ̂ = ϕ(t − a) = e−iaω fˆ(ω) and thus
V
hF {τa T } , ϕi = hT , e−iaω ϕ̂i
Finally, since ω 7→ e−iωa is a tempered function, e−iaω T is a tempered distribution
and satisfies:
5.10 Properties of the Fourier transform in S 0
97
he−iωa T , ϕ̂i = hT , e−iaω ϕ̂i = hF {τa T } , ϕi
Example 5.14 — Fourier transform of a displaced Dirac. The Fourier transform
of the Dirac distribution at a point a is:
F {δa } = F {τa δ}
and using the property of translation 5.30
1
F {δa } = e−iωa F {δ} = √ e−iωa 1
2π
Theorem 5.10.2 — Fourier transform and modulation. Let T ∈ S 0 (R; C). Then,
AF
n
T
for any a ∈ R
o
F eiat T = F {τa T }
a∈R
(5.31)
DR
Proof
By using the definition of the Fourier transform and the property (5.19):
n
o
hF eiat T , ϕi = heiaω T , ϕ̂i = hT , eiaω ϕ̂i
ϕ ∈ S(R; C)
V
Now, since eiaω ϕ̂i = ϕ(t + a)
n
o
V
hF eiat T , ϕi = hT , ϕ(t + a)i = hF {T } , τ−a ϕi = hτa F {T } , ϕi
Theorem 5.10.3 — Fourier transform and dilation-contraction. Let T ∈ S 0 (R; C).
Then, for any a ∈ R {0}
F {∂a T } =
1
∂1/a F {T } a ∈ R {0}
|a|
(5.32)
Proof
Using the property (5.14) and the definition of the Fourier transform in S 0 one
gets:
hF {∂a T } , ϕi = h∂a T , ϕ̂i = hT ,
1
∂1/a ϕ̂i
|a|
∀ ϕ ∈ S(R; C)
Fourier transform of tempered distributions
98
and using the dilation/contraction property of integrable functions 1/|a|∂1/a ϕ̂ =
∂a ϕ
V
V
hF {∂a T } , ϕi = hT , ∂a ϕi = hF {T } , ∂a ϕi = h
1
∂1/a F {T } , ϕi
|a|
∀ ϕ ∈ S(R; C)
Example 5.15 — Fourier transform of sinω0 t. Let us consider a sinusoid of fre-
quency ω0 > 0, sin(ω0 t). It defines a distribution that correspods to sin(ω0 t) =
1
∂
sin. Its Fourier transform is, according to 5.32:
ω0 1/ω0
√
√
1
2π
2π
F
∂1/ω0 sin = ∂ω0 F {sin} =
∂ω0 (δ1 − δ−1 ) =
(δω0 − δ−ω0 )
ω0
2i
2i
AF
T
so that its Fourier transform has two Dirac distributions at ±ω0 :
√
2π
(δω0 − δ−ω0 )
F {sin(ω0 t)} =
2i
Theorem 5.10.4 — Fourier transform and differentiation. Let T ∈ S 0 (R; C).
DR
Then, for any α ∈ N
F {Dα T } = (iω)α F {T }
α∈N
(5.33)
Proof
By applying α times the property 5.15 and (2.6) for integrable functions .
Example 5.16 — Fourier transform of the Heavyside distribution. Here is another
example of a function that only has a Fourier transform in the framework of
distributions. The Heavyside distribution H is such that
H0 = δ
so that 5.33 gives:
F {H 0 } = F {δ} = iωF {H}
and since F {δ} =
√1 1:
2π
√
2πiωF {H} = 1
5.10 Properties of the Fourier transform in S 0
99
recalling that ωδ = 0, and that ω v.p ω1 = 1, the general solution to the previous
identity is:
1
1
v.p + Aδ
F {H} = √
ω
2πi
where A is a constant.
Theorem 5.10.5 — Fourier transform and multiplication by t. Let T ∈ S 0 (R; C).
Then, for any α ∈ N
F {(−it)α T } = Dα F {T } α ∈ N
(5.34)
Proof
By applying α times the property 5.19 and (2.5) for integrable functions .
T
Fourier transform in E 0 (R)
AF
For every distribution with compact support T ∈ E 0 (R), the tempered distribution
F{T} is the distribution defined by the function ω 7→ FT (ω)
1
ω ∈ R 7→ √ hT, e−iωt i
2π
the function FT (ω) is of class C ∞ (R; C) and of polynomial growth, as well as for
all of its derivatives.
DR
5.10.1
Proof
we have
−iωt
hhT, e
i, ϕi =
Z
hT, e−iωt iϕ(ω)dω
R
= hT,
Z
e−iωt ϕ(ω)dωi =
√
2πhT, ϕ̂i =
√
2πhF{T}, ϕi
R
Theorem 5.10.6 — Fourier transform and convolution.
• a) Let T ∈ S 0 (R; C)
and X ∈ S(R; C). Then:
F {X ∗ T } =
√
2π X̂F {T }
(5.35)
• b) Let T ∈ S 0 (R; C) and S ∈ E 0 (R; C), then
F {S ∗ T } =
√
2πFSF {T }
∈ S 0 (R)
(5.36)
Fourier transform of tempered distributions
100
where FS was defined in 5.10.1.
Proof
a) One has
hF {X ∗ T } , ϕi = hX ∗ T , ϕ̂i = hT , ∂−1 (X ∗ ϕ)i
V
∀ ϕ ∈ S(R; C)
√
And according to the inversion formula and (2.10), ∂−1 X ∗ ϕ = X ∗ ϕ = 2π X̂ϕ
V
V
V
V
V
√
√
√
hF {X ∗ T } , ϕi = hT , 2π X̂ϕi = hF {T } , 2π X̂ϕi = h 2π X̂F {T } , ϕi
V
∀ ϕ ∈ S(R; C)
T
b) Since FS is a function of class C ∞ (R; C) with polynomial growth, as well
as all of its derivatives, and F {T } ∈ S 0 (R), it follows that FSF {T } ∈ S 0 (R) .
Now, for every ϕ ∈ S(R; C)
AF
hF(S ∗ T), ϕi = hS ∗ T, ϕ̂i = hS ∗ T, F −1 (∂−1 ϕ)i
= hT, (∂−1 S) ∗ F −1 (∂−1 ϕ)i = hT, ∂−1 (S ∗ F −1 ϕ)i
= hT, FF(S ∗ F −1 ϕ)i = hF{T}, F(S ∗ F −1 ϕ)i
(5.37)
DR
Now, writing ψ = F −1 ϕ ∈ S(R; C), and according to (8.9) S ∗ ψ ∈ S(R; C) and
we have
Z
1 Z −iωt0
1
0
e
F(S∗F −1 ϕ)(ω) = F(S∗ψ)(ω) = √ √
hS, ψ(t0 −·)idt0 = hS, e−iωt ψ(t0 − t)dt0 i
2π 2π R
|R
{z
}
ψ̂(ω)e−iωt
F(S ∗ F −1 ϕ)(ω) = ψ̂(ω)hS, e−iωt i = ψ̂(ω)
√
2πFS(ω)
| {z }
=ϕ(ω)
and so
hF(S ∗ T), ϕi = hF{T}, F(S ∗ F −1 ϕ)i
= hF{T}, FSϕi = hFSF{T}, ϕi
and we have shown that
F(S ∗ T) = FSF{T}
(5.38)
5.10 Properties of the Fourier transform in S 0
101
Theorem 5.10.7 — Fourier transform and multiplication. Let T ∈ S 0 (R; C) and
X ∈ S(R; C). Then:
1
F {XT } = √ X̂ ∗ F {T }
2π
(5.39)
Proof
One has
hF {XT } , ϕi = hXT , ϕ̂i = hT , X ϕ̂i
∀ ϕ ∈ S(R; C)
√
and according to (2.12) X ϕ̂ = (∂−1 X )ϕ̂ = 1/ 2π∂−1 X̂ ∗ ϕ
V
V
V
1
1
hF {XT } , ϕi = hT , √ ∂−1 X̂ ∗ ϕi = h √ F {T } , ∂−1 X̂ ∗ ϕi
2π
2π
V
T
Finally
AF
1
hF {XT } , ϕi = h √ X̂ ∗ F {T } , ϕi
2π
The Dirac’s comb
∀ ϕ ∈ S(R; C)
DR
5.10.2
∀ ϕ ∈ S(R; C)
Let a ∈ R>0 . The Dirac’s comb of period a is defined as:
comba =
∞
X
(5.40)
δna
n=−∞
so that:
hcomba , ϕi =
∞
X
ϕ(na)
∀ ϕ ∈ S(R; C)
(5.41)
n=−∞
Remark
• The sum in Eq.(5.41) is finite for every ϕ ∈ S(R; C), so there is no problem
of convergence and the distribution comba is well defined.
• The product of an infinitely differentiable function f ∈ C ∞ (R; C), which is a
tempered function, and comba is well-defined and
hf comba , ϕi = hcomba , f ϕi =
∞
X
n=−∞
so that
f (na)ϕ(na)
∀ ϕ ∈ S(R; C)
Fourier transform of tempered distributions
102
f comba =
∞
X
f (na)δna
(5.42)
n=−∞
which represents a "sampling" of the function f at regular intervals spaced
by a.
Example 5.17 — Fourier transform of the Dirac’s comb (Poisson’s formula).
One has
∞
X
X
1 +∞
√
e−iωna 1
F {comba } =
F {δna } =
2π n=−∞
n=−∞
T
so that the Fourier transform of comba is a periodic distribution of period
2π/a. Let us then calculate its restriction to [−π/a, π/a]. For this, consider
ϕ ∈ S(R; C) such that ϕ is zero outside [−π/a, π/a]. Now, the sequence CN =
P+N
−iωna
√1
1 is such that CN → F {comba }. One has:
n=−N e
2π
AF
+N
1 Z π/a sin((N + 1/2)aω)
1 Z X
hCN , ϕi = √
e−iωna ϕ(ω)dω = √
ϕ(ω)dω
sin(aω/2)
2π R n=−N
2π −π/a
where we have used the geometric series
aω/2
ϕ(ω) sin(aω/2)
P+N
n=−N
e−inaω =
sin((N +1/2)aω)
.
sin(aω/2)
Now let
2
DR
us define ψ̂(ω) =
∈ L (R; C) since it vanishes outside [−π/a, π/a]. It
is the Fourier transform of a square integrable function ψ and√therefore we can
apply the Plancherel-Parseval’s theorem (2.7.1) recalling that 1/ 2πSinc(ωu/2) is
the Fourier transform of uu :
1 Z sin((N + 1/2)aω)
1 Z (N +1/2)a
hCn , ϕi = √
ψ̂(ω)dω =
ψ(t)dt
aω/2
a −(N +1/2)a
2π R
and
√
lim hCN , ϕi = hF {comba } , ϕi =
N →∞
2π
ψ̂(0) =
a
√
√
2π
2π
ϕ(0) = h
δ, ϕi
a
a
Finally, since F {comba } is 2π/a-periodic, we obtain that the Fourier transform
of a Dirac’s comb is also a Dirac’s comb:
F {comba } = F
( +∞
X
n=−∞
)
δna
√
√
X
2π +∞
2π
=
δ2πk/a =
comb2π/a (5.43)
a k=−∞
a
√
in nparticular ofor a = 2π one gets an eigenfunction of the Fourier transform
F comb√2π = comb√2π (a property shared with the gaussian function of unit
width). Eq.(5.43) can be rewritten as (Poisson’s formula):
5.11 Sampling
103
X
1 +∞
√
e−iωna 1 =
2π n=−∞
√
X
2π +∞
δ2πk/a
a k=−∞
(5.44)
Sampling
Eq.(5.42) represents a uniform sampling of a function f :
fd = f comba =
∞
X
f (na)δna
(5.45)
n=−∞
T
Let us see what is the Fourier transform of fd . If {f (na)}n∈Z is bounded, fd is a
tempered distribution and its Fourier transform is well-defined. One obtains:
AF
1
F {fd } = √ fˆ ∗ F {comba }
2π
and using (5.43):
√

∞
1 ˆ  2π X
δ2πk/a 
F {fd } = √ f ∗
a k=−∞
2π
DR
5.11
and since fˆ ∗ δ2πk/a = τ2πk/a fˆ1
F {fd } =
X
1 +∞
2kπ
fˆ(ω −
)
a k=−∞
a
(5.46)
Sampling a signal is therefore equivalent to a "periodisation" of its Fourier transform.
Theorem 5.11.1 — Nyquist theorem. Let f be a signal whose Fourier transform
fˆ vanish outside [−π/a, π/a]. Then f can be reconstructed by an interpolation
of its samplings
f (t) =
+∞
X
f (nT )hT (t − nT )
n=−∞
with
h(t) = Sinc
sin πt
πt
= πt a
a
a
Fourier transform of tempered distributions
104
Proof
Since fˆ(ω) vanish outside [−π/a, π/a], fˆ(ω − 2nπ/a) does not intersect fˆ(ω) and
one has, according to (5.46):
fˆ(ω)
fˆd (ω) =
a
π
a
|ω| ≤
This can be rewritten as:
fˆ(ω) = 2π fˆd (ω) u2π/a (ω)
so that
f (t) = fd (t) ∗ ha = ha ∗
∞
X
f (na)ha (t − na)
n=−∞
DR
AF
T
n=−∞
∞
X
f (na)δna =
5.12 Summary
Summary
• The Fourier transform F is a linear continuous bijection
F : S 0 (R; C) 7→ S 0 (R; C)
defined by:
hF {T } , ϕi = hT , ϕ̂i
∀ϕ ∈ S(R; C)
• this is a generalization of the Fourier transform in L1 and L2 . Indeed, any
p-integrable function f defines a tempered distribution and one has:
F {f } = F {f }
• Translation
a∈R
T
F {τa T } = e−iωa F {T }
AF
• Modulation
n
o
F eiat T = F {τa T } a ∈ R
• Dilation / Contraction
F {∂a T } =
1
∂1/a F {T }
|a|
DR
5.12
105
a ∈ R {0}
• Differentiation
F {Dα T } = (iω)α F {T }
α∈N
• Multiplication by t
F {(−it)α T } = Dα F {T } α ∈ N
• Convolution
For X ∈ S(R; C):
F {X ∗ T } =
√
2π X̂F {T }
• Multiplication
For X ∈ S(R; C):
1
F {XT } = √ X̂ ∗ F {T }
2π
• Inverse
∀ T ∈ S 0 (R; C),
F {F {T }} = ∂−1 T
Fourier transform of tempered distributions
106
• Some Fourier transforms
AF
DR
where a ∈ R.
T
1
δ ←→ √ 1
2π
√
2π
sin(ω0 t) ←→
(δω0 − δ−ω0 )
2i
!
1
1
i
v.p + δ
H ←→ √
ω 2
2πi
ωa 1
√
Sinc
ua (t) ←→
2
2π
1
1
e−at 1[R+ ] ←→ √
2π a + iω
2a
1
e−a|t| ←→ √
2π a2 + ω 2
1
1
2
2
√
e−t /(2a) ←→ √ e−aω /2
2πa
2π
ωa 1
∧a (t) ←→ √ Sinc2
2
2π
DR
AF
T
6 — Fourier series
Fourier series
108
6.1
Introduction
Usually, the Fourier transform and the Fourier series are defined for integrable (or
square-integrable) and periodic functions, respectively. Since no periodic function
(except from zero) belongs to L1 or L2 , the Fourier transform and the Fourier
series appear as two disjoint theories. We will see in this chapter how both are
unified in the more general framework of distributions.
Fourier series of periodic functions
Let f : R 7→ C be a T -periodic function, that is f (x + T ) = f (x) for every
x ∈ R. One defines the vector spaces L1T and L2T of T −periodic functions that are
integrable and square-integrable over [0, T ], respectively. One has L2T ⊂ L1T . In
L2T , we can define the inner product as:
1 Z a+T
1ZT
f (t)∗ g(t)dt =
f (t)∗ g(t)dt
T 0
T a
q
(f |f ) =
√1
T
R
T
0
1/2
∀a ∈ R
T
and the norm as kf kL2T =
f (t)dt
.
AF
(f |g) =
Theorem 6.2.1 — Fourier series in L2T . (a) every f ∈ L2T can be uniquely decom-
posed as the following series:
Sf (t) =
X
cn (f )einω0 t
(6.1)
DR
6.2
n∈Z
where ω0 = 2π/T and the coefficients cn (f ) are given by
cn (f ) = (einω0 t |f ) =
1ZT
f (t)e−inω0 t dt
T 0
(6.2)
the series Sf converges to f in L2T , this means:


X
1ZT

lim
|f −
cn (f )einω0 t |2  = 0
M,N →∞ T 0
n∈Z
(b) one has the Bessel-Parseval identity:
kf k2L2 =
T
X
1ZT
|f (t)|2 dt =
|cn (f )|2
T 0
n∈Z
In addition, if g ∈ L2T :
(6.3)
6.2 Fourier series of periodic functions
X
cn (f )cn (g)∗ =
n∈Z
109
1ZT
f (t)g(t)∗ dt
T 0
(b) reciprocally, if the coefficients (γn )n∈Z are such that n∈Z |γn |2 < ∞, the
P
series n∈Z γk einω0 t converges in L2T to a function f ∈ L2T and f is the only
element of L2T such that cn (f ) = γn .
P
Remarks
• One always has
1ZT
1
|ck (f )| ≤
|f (t)|dx = kf kL1T ≤ supt f (t)
T 0
T
2
• The convergence of the series in LT does not imply that Sf (t) converges to
f (t) for almost every value of t. It does not mean that for each value of t, the
complex number f (t) is equal to the sum of the series on the right. Clearly,
if f is modified at a single point or in any ensemble of zero measure, the
series will remain unchanged since the Fourier coefficients are not modified.
Rather, the series converges under the L2T norm to f
0
AF
Z T
T
2
|f −
X
cn (f )einω0 t |2 dt = 0
n∈Z
DR
• The coefficients (6.2) are well defined if f ∈ L1T . This is always the case for
f ∈ L2T ⊂ L1T .
• It can be convenient to rewrite the Fourier series by puting together the
terms with opposite values of k and to obtain a series of cosinus and sinus:
f=
∞
a0 X
+
an cos nω0 t + bn sin nω0 t
2
n=1
(6.4)
with a0 = 2c0 and
an = cn +c−n =
2ZT
f (t) cos nω0 tdt,
T 0
bn = i(cn −c−n ) =
2ZT
f (t) sin nω0 tdt
T 0
specially when f is real (an and bn are therefore real) or if f has parity
properties (then one obtains a series of sinus or cosinus). In this notation
the Bessel-Parseval identity is written:
kf k2L2 =
T
∞
X
1ZT
1
(|an |2 + |bn |2 )
|f (t)|2 dt = |a0 |2 +
T 0
4
2
n=1
Theorem 6.2.2 If f ∈ L2T and its Fourier coefficients satisfy
X
n∈Z
|cn (f )| < ∞
Fourier series
110
then f is equal to a continuous function g almost everywhere and the Fourier
series of f converges uniformly to g on R.
Theorem 6.2.3 — Riemann-Lebesgue for the Fourier coefficients. Let f ∈ L1T ,
its Fourier coefficients given by (6.2) verify
|cn (f )| ≤
1ZT
|f (t)|dt
T 0
and cn (f ) decay to 0 as |n| → ∞
T
Remark
Note that the condition f ∈ L1T is less restrictive than f ∈ L2T and the Fourier
coefficients (6.2) are still well-defined. One could expect to obtain a convergence in
L1 or almost everywhere of the Fourier series (6.1) for f ∈ L1T , but this is wrong.
It can be shown that there exists a function f ∈ L1T whose Fourier series diverges
at every point! This catastrophe will be solved in the framework of tempered
distributions (to be discussed later). The following theorem gives additional conditions for which there is a certain convergence of the Fourier series of f ∈ L1T .
Definition 6.2.1 — Functions of bounded variation. For every finite subdivision
AF
σ = (t0 = 0, t1 , ..., tn = b) of the interval [a, b] one defines
V (f, σ) =
n−1
X
|f (xi+1 ) − f (xi )|
DR
i=0
and the total variation of f over the interval [a, b] is, by definition
V (f ; [a, b]) = sup V (f, σ)
(6.5)
σ
• If V (f ; (a, b)) is finite, f is said to be of bounded variation.
• Every monotonic, bounded function is of finite variation and V (f ; (a, b)) =
|f (b) − f (a)|
• A continuous function with bounded derivative is of bounded variation if
(a, b) is bounded and
V (f ; (a, b)) =
Z b
|f 0 (t)|dt
a
• If f has a finite number of discontinuities of first kind and if in all the
other points f has a derivative bounded in modulus by a fixed number, f
6.2 Fourier series of periodic functions
111
is of bounded variation.
• A function f of bounded variation is not necessarily continuous, but it
has at every point t a left-handed limit and a right-handed limit:
f (t − 0) = lim f (x − )
→0,>0
f (t + 0) = lim f (x + )
→0,>0
these two limits are equal (f is therefore continuous) except in a countable
set of points.
• There are differentiable functions which do not have bounded variation.
For example, f : [−1, 1] 7→ f (t) = t2 cos2 (π/t2 ) if t 6= 0 and f (t) = 0 if
t = 0.
Theorem 6.2.4 — Dirichlet-Jordan theorem. Let f be a function of bounded
variation in the interval [0, T ]. The Fourier series of f
X
cn (f )einω0 t
n∈Z
N →∞
n=+N
X
n=−N
cn (f )eikω0 t =
1
(f (t − 0) + f (t + 0))
2
(6.6)
DR
lim
AF
T
is semi-convergent at every point t to the arithmetic average between the lefthand limit f (t − 0) and the righ-hand limit f (t + 0). The semi-convergence
P
ikω0 t
means that the symmetric series fN (t) = n=+N
is such that
n=−N cn (f )e
If, in addition, f is continuous at t = t0 , then fN (t0 ) → f (t0 ). Finally, if f is
continuous for every point t in the interval [0, T ] (therefore continuous in 0 and
T ) this semi-convergence of the series is uniform in [0, T ]
Theorem 6.2.5 Let f be a T -periodic function of class C 1 piecewise, and let ck
be its Fourier coefficients.
(a) At every point t where f is continuous, the symmetric, partial sums SN (t) =
PN
ikωt
converge to f (t).
−N ck e
(b) At every point t where f is discontinuous, SN (t) converge to (f (t + 0) −
f (t − 0))/2.
(c) Let [a, b] be an interval such that f is continuous at every point of [a, b].
Then SN converges uniformly to f over [a, b].
Remark
If we only know that f is continuous, there is in general no uniform convergence,
Fourier series
112
and not even simple convergne: there is indeed a continuous function such that its
Fourier series diverges at every point t of a finite, uncountable set.
Proof
One has cn (f ) =
RT
1
T
0
cn (f ) =
f (t)e−iω0 nt dt and f 0 ∈ L2T . Integration by parts gives:
T
1
1 ZT 0
1
f (t)e−iω0 nt dt
[−
f (t)e−iω0 nt ] +
T ω0 n
T
iω
n
0
0
0
which prooves that
cn (f ) =
1
cn (f 0 )
iω0 n
and one has
1
1 ZT 0
|f (t)|dt
|cn (f 0 )| ≤
ω0 |n|
w|n|T 0
T
|cn (f )| =
(c) Finally, note that n∈Z |cn (f )| < ∞ and this implies that fN is a Cauchy
sequence in the uniform norm on [0, T ] (and hence on R), so fN converges uniformly on R to some continuous function g. Since uniform convergence implies
convergence in the L2T norm, it follows that f = g almost everywhere. But both f
and g are continuous, so f = g everywhere.
DR
AF
P
Theorem 6.2.6 (a) Let f be a T −periodic function of class C k . Then
cp
dk f
dtk
!
= (iω0 p)k cp (f )
(6.7)
there exists a constant C such that |cp (f )| ≤ C/|p|k for k 6= 0.
(b) If f is of class C 2 , its Fourier series converges to f uniformly.
Proof
One has
|cp (f )| ≤
1 kZ T k
C
|f (t)|dt = k
T pω0 0
|p|
In particular, for k = 2, |cn (f )| is bounded by C/k 2 , and therefore the series
P
|c (f )einω0 t | is bounded, independently of t, by the convergent series |c0 | +
Pn∈Z n C
n∈Z,n6=0 p2 . This proves that the fourier series of f converges absolutely and
uniformly to a continous function g. Since f is continuous, f = g.
6.3 Periodic distributions
Periodic distributions
Let us recall Poisson’s summation formula (5.44)
X
1 +∞
√
e−iuna 1 =
2π n=−∞
√
X
2π +∞
δ2πk/a
a k=−∞
where u is the independent variable of S 0 (R; C). By using the substitution 2π/a = T
and calling t the independent variable of S 0 , it can be rewritten as:
+∞
X
X
1 +∞
inω0 t
e
1=
δkT
T n=−∞
k=−∞
T
with ω0 = 2π/T . One recognizes the Fourier series decomposition of the Dirac’s
comb of period T . In the following we will define the Fourier series for any periodic
distribution.
Definition 6.3.1 — Space of T -periodic distributions. Let PT be the set of C ∞
functions that are T −periodic. One defines the space of T -periodic distributions
PT0 as:
AF
PT0 := {u : PT → C, u is linear and continuous}
and we write, for every ϕ ∈ PT0 , u(ϕ) = hu, ϕiT
Example 6.1 If f ∈ L1T , it defines a periodic distribution f by
DR
6.3
113
hf , ϕiT =
Z T
f (t)ϕ(t)dt
∀ϕ ∈ mathcalP 0
0
Theorem 6.3.1 — Fourier series in PT0 . If u is a T -periodic distribution, its
Fourier coefficients are defined as:
ck (u) =
1
hu, e−ikω0 t iT
T
k∈Z
and one has
u=
X
ck (u)eikω0 t
k∈Z
the convergence of the series means:
lim h
N →∞
X
ck (u)eikω0 t , ϕiT = hu, ϕiT
∀ϕ ∈ PT
|k|≤N
Let f ∈ L1T and F the T -periodic funcion over R associated with f . To every
ϕ ∈ S(R; C) one can build a T -periodic function Φ by
Fourier series
114
Φ(t) =
X
∈ PT
ϕ(t + kT )
k∈Z
and one has
Z
Z
f (t)Φ(t)dt =
T
F (t)ϕ(t)dt
R
Indeed, the right hand term is:
Z
F (t)ϕ(t)dt =
X Z (k+1)T
F (t)ϕ(t)dt =
k∈Z kT
R
XZ T
F (t)ϕ(t + kt)dt =
k∈Z 0
Z T
F (t)Φ(t)dt
0
To every T-periodic distribution u ∈ PT0 one can associate the T −periodic distribution over R, U ∈ S 0 (R; C) by generalization of the former identity
hU , ϕi = hu, ΦiT
AF
T
indeed U is T −periodic since replacing ϕ by ϕ(t + T ) does not change the value
of Φ(t). For example, if u = δ over [−T /2, T /2], one has:
hU , ϕi =
X
ϕ(kT ) =
k∈Z
X
δkT
k∈Z
DR
which is the Dirac’s comb. More generally
hU , ϕi =
X
hu, ϕ(t + kT )iT =
k∈Z
U=
X
hτkT u, ϕi ∀ϕ ∈ S(R; C)
k∈Z
X
τkT u
k∈Z
Reciprocally, every T −periodic distribution over R U is associated with a unique
u ∈ PT0 .
Theorem 6.3.2 — Fourier series in S 0 . Every T −periodic distribution U can be
uniquely written as a convergent Fourier series in S 0 (R; C)
U=
X
cn (U )einω0 t
(6.8)
n∈Z
where ω0 = 2π/T and the series cn (U ) is slowly increasing. The Fourier
coefficients can be calculated by
cn (U ) =
1
hu, e−inω0 t i
T
(6.9)
6.4 Link between the Fourier series and the Fourier transform
by choosing a distribution u ∈ PT0 such that U =
115
P
k∈Z τkT u
Proof
Indeed, one has:
Link between the Fourier series and the Fourier transform
Any T −periodic distribution U ∈ S 0 (R; C) can be thus written as a Fourier series
U=
X
cn (U )einω0 t 1
n∈Z
In addition, U has a well-defined Fourier transform and:
F{U } =
X
cn (U )F{einω0 t 1} =
n∈Z
X
√
cn (U ) 2πδnω0
n∈Z
AF
T
so that the Fourier transform F{U } is a series
of Diracs centered at nω0 , n ∈ Z,
√
each one scaled by the Fourier’s coefficient 2πcn (U ). Both the Fourier series and
the Fourier transform are strictly related in the sense of distributions.
DR
6.4
T
AF
DR
DR
AF
T
7 — Laplace Transform
Laplace Transform
118
7.1
Introduction
In Fourier Analysis, the Fourier basis are complex exponentials of a real variable
ω. This has the advantage of interpreting ω as frequency, and the whole Fourier
Transform as a spectrum. However, as we have seen throughout this book, the
convergence of the Fourier integral is not guaranteed for functions outside L1 . One
possible way to overcome this limitation is to consider a complex variable s instead
of the real variable ω in the Fourier basis, where s = σ + iω.
This change of variables means that the basis for this modified Fourier transformation would be now:
e−st = e−σt e−iω
Laplace Transform
Let f : R 7→ C be a causal function, that is, f = 0 for t < 0. The Laplace
transform L{f } of f is the application C 7→ C defined by:
L{f }(s) =
DR
7.2
AF
T
For certain values of σ, the term e−σt forces a function to decay exponentially
to zero at t → ∞. This idea gives birth to the Laplace Transform, which can
be applied in general to locally integrable functions (which are not necessarily
integrable over R). All the functions admitting a Fourier transform in L1 (R; C)
admit a Laplace transform, but the reciprocal does not hold.
Z
R+
f (t)e−st dt
s∈C
(7.1)
The Laplace Transform is widely used for initial-value problems, such as transients
on dynamical systems, but it is not so physically interpretable as the Fourier
Transform is. Note that the existence of L{f }(s) imposes that
t 7→ |f (t)e−st | = |f (t)|e−Re{s}t
∈ L1 (R+ ; C)
Let us define the minimal abcissa of convergence as the lower bound of σ ∈ R such
that f (t)e−σt is integrable:
α = inf{σ ∈ R | t 7→ f (t)e−σt ∈ L1 (R+ ; C)}
then, the Laplace transform of f is defined for Re{s} > α. For Re{s} = α, the
integral in Eq.(7.1) may or not converge.
Theorem 7.2.1 — Existence of the Laplace integral. (a) If f is locally integrable,
that is f ∈ L1loc (R+ ; C), then there exists a minimal abcissa of convergence α
such that −∞ ≤ α < ∞.
7.2 Laplace Transform
119
(b) If f does not grow faster than any exponential at infinity, that is if
|f (t)| ≤ Keat for t ≥ M , for any real constant a and positive constants K, M .
This implies that the minimal abcissa of convergence α is such that
α≤a
and the Laplace Transform L{f }(s) exists for Re{s} > a.
Theorem 7.2.2 — Region of convergence. If f is a function of minimal abcissa
of convergence α, then its Laplace transform L{f } exists in the open half-plane
Re{s} > α, also called region of convergence of L{f }. In addition, L{f } is
bounded and holomorphic in the region of convergence (i.e, it has a complex
derivative).
R+
−Re{s}t
|f (t)|e
dt <
Z
|f (t)|e−αt dt < +∞
AF
|L{f }(s)| ≤
Z
T
Proof
Indeed, for Re{s} > α, the Laplace transform exists and is bounded:
R+
7.2.1
DR
The demonstration of the Laplace transform being holomorphic in the region of
convergence will be left for later.
Link between Laplace and Fourier transforms
Let f : R+ 7→ C be a causal, locally integrable function with a minimal abcissa
of convergence α < ∞. Then, the function t 7→ f (t)e−σt is integrable over R for
σ > α, and therefore has a Fourier transform in L1 given by
1 Z
1
F{f (t)e−σt }(ω) = √
f (t)e−σt e−iωt dt = √ L{f }(σ + iω)
2π R
2π
and so for σ > α
L{f }(σ + iω) =
√
2πF{f (t)e−σt }(ω)
(7.2)
The interpretation is as follows: the Laplace transform of f evaluated at s ∈ C is
the Fourier transform of f evaluated at ω = Im{s} when the f is forced to decay
to zero at infinity as f (t)e−Re{s}t .
Example 7.1 — Laplace transform of the heaviside step function. Consider the
heaviside function H(t). Its minimal abcissa of convergence is α = 0, since
t 7→ H(t)e−σt is integrable over R+ for σ > 0. Its Laplace transform is given by:
Laplace Transform
120
L{H(t)}(s) =
Z
R+
e−st dt =
1
s
Re{s} > 0
Example 7.2 — Laplace transform of a power of t. Consider the function f :
+
R → (0, ∞) given by
f (t) = tn
n∈N
which is a typical example of a non-integrable function over R whose Fourier
transform does not exist outside the framework of distributions. Its Laplace
transform, however, is well defined for Re{s} > 0. Indeed, t → te−σt is integrable
over R for σ > 0, and so the minimum abscissa of convergence for f is α = 0. Let
us begin with the simple case n = 1 (f (t) = t). Integrating by parts, we obtain:
L{t} =
Z
R+
te−st dt =
1
1Z
e−st dt = 2 Re{s} > 0
s R+
s
L{t } =
Z
R+
n −st
t e
nZ
n!
dt =
tn−1 e−st dt = n+1 Re{s} > 0
s R+
s
AF
n
T
and by recurrence, we obtain for f (t) = tn , n ∈ N
DR
the particular case n = 0 corresponds to the Heaviside function whose Laplace
transform is, indeed, 1/s.
Example 7.3 — Laplace transform of an exponential. Consider an exponential
function t → eat with a ∈ R. This function is not integrable for a > 0. Its Laplace
transform is well defined for Re{s} > a:
L{eat }(s) =
Z
R+
eat e−st dt =
1
Re{s} > a
s−a
√
Example 7.4 — Laplace transform of 1/ t. Consider the function f : R+ →
(0, ∞) given by
1
f (t) = √
t
t>0
Its Laplace transform exists for Re{s} > 0 since t−1/2 e−σt ∼ e−σt when t → ∞.
We have
Z
Z
1
1
2
√ e−st dt = 2
L{ √ }(s) =
e−su du =
R+
R+
t
t
where we have used
r
π
Re{s} > 0
s
7.3 Properties of the Laplace transform
Z
2
−su2
R+
e
du =
Z
121
2
e−su du = I
R
and
I2 =
Z Z
R
e−s(x
2 +y 2 )
dxdy = 2π
Z ∞
2
e−sr rdr =
0
R
π
s
Example 7.5 — Laplace transform of the Bessel function of the 1st kind. Con-
sider the Bessel function of the first kind of order n ∈ N
1 Z π i(nθ−t sin θ)
Jn (t) =
e
dθ
2π −π
and consider the causal function f (t) = H(t)Jn (t). Its Laplace transform reads
T
Z ∞
1 Z ∞ Z π i(nθ−t sin θ) −st
1 Z π inθ
−(s+i sin θ)t
e
dt dθ
L{Jn (t)H(t)}(s) =
e
dθe dt =
e
2π 0 −π
2π −π
0
AF
For Re{s} > 0
1 Z π inθ
1
1 Zπ
einθ
L{Jn (t)H(t)}(s) =
e
dθ =
−iθ dθ
2π −π
s + i sin θ
2π −π s + eiθ −e
2
DR
consider the substitution z = eiθ , so that dz = idθeiθ , the last integral becomes an
integral over the unit circle γ in the complex plane in the clockwise direction
L{Jn (t)H(t)}(s) =
1 I
z n−1
1 I
z n−1
dz
=
2z
dz
2πi γ s + z−z2 −1
2πi γ 2sz + z 2 − 1
L{Jn (t)H(t)}(s) =
1 I
2z n
dz
2πi γ (z − z0 )(z − z1 )
where z0 and z1 are the roots of the polynomial z 2 + 2sz − 1, given by
z0 =
√
s2 + 1 − s
√
z0 = − s2 + 1 − s
7.3
Properties of the Laplace transform
When studying the properties of the Laplace transform, we will encounter remarkable symmetries with the properties of the Fourier transform fˆ. For example,
translating f corresponds to a multiplication of L{f } by an exponential, has it
also happens with the Fourier Transform
Laplace Transform
122
7.3.1
Laplace transform and translation/modulation
Theorem 7.3.1 Let f : R 7→ C a causal function and iL{f } its Laplace transform.
Then, for any a ∈ R+ and b ∈ C
L{f (t−a)H(t−a)}(s) = e−as L{f }(s)
and
L{e−bt f (t)} = L{f }(s+b) (7.3)
Example 7.6 — Laplace transform and modulation. Consider t 7→ tn eat . Its
Laplace transform is given by
L{tn eat }(s) = L{tn }(s − a) =
n!
(s − a)n+1
similarly, the Laplace transform of t 7→ e−at sin at can be obtained
L{e−at sin at}(s) = L{sin at}(s + a) =
a
(s + a)2 + a2
Example 7.7 — Laplace transform of
r
π
s
t. Recalling that
AF
H(t)
L{ √ }(s) =
t
√
T
H(t)e±it /
Re{s} > 0
the property of modulation gives
H(t)
√ }(s) = L{ √ }(s ∓ i) =
t
t
L{e
7.3.2
DR
±it H(t)
π
s∓i
Re{s} > 0
Laplace transform and dilation/contraction
Theorem 7.3.2 Let f be a causal function and L{f } its Laplace transform. For
1
a ∈ R/ {0} we define fa (t) = |a|
f (t/a). Then
L{fa (t)}(s) = L{
7.3.3
s
1
f (t/a)}(s) = L{f (t)}(as)
|a|
(7.4)
Laplace transform and differentiation
Theorem 7.3.3
• Suppose a causal function f with minimal abscissa of convergence α. Then, the function t ∈ R+ 7→ (−t)m f (t) has the same minimal
abcissa of convergence. Moreover, the Laplace transform L{f } is homolorphic in the plane Re{s} > α and
dm L{f }(s) Z
=
(−t)m f (t)e−st dt = L{(−t)m f }(s)
m
+
ds
R
(7.5)
7.3 Properties of the Laplace transform
123
Indeed, differentiation under the integral sign is justified since for every s
such that Re{s} > α
|(−t)m f (t)e−st | ≤ tm |f (t)|e−αt ∈ L1 (R, C)
• Let f be a function of class C 1 (R; C) whose Laplace transforms has a
minimal abcissa of convergence α. If, in addition, for Re{s} > α the
function t 7→ e−st f (t) admits a limit when t → ∞, then the function f 0
has a Laplace transform with abcissa of convergence equal to α and
L{f 0 (t)}(s) = sL{f }(s) − f (0+ )
(7.6)
and this result can be generalized for a n order derivative, where the
continuity of the first (n − 1) derivatives at t 6= 0 is required
L{
Example 7.8 — Laplace transform of te−at and t sin t. Consider the function
T
dn f
dn−1 f +
n
n−1
+
n−2 df
+
}(s)
=
s
L{f
}(s)−s
f
(0
)−s
(0
)−...−
(0 ) (7.7)
dtn
ds
dsn−1
−at
L{te
AF
f (t) = te−at . Its Laplace transform can be easily obtained using 7.5
d
1
d
}(s) = − L{e−at }(s) = −
ds
ds s + a
Re{s} > a
DR
In the same way, the Laplace transform of g(t) = t sin t is given by
L{t sin t}(s) = −
d
1
2s
L{ 2
}(s) = 2
ds s + 1
(s + 1)2
Re{s} > a
Example 7.9 — Laplace transform of sin tH(t). Consider the causal function
f (t) = sin tH(t). Its Laplace transform may be easily obtained by considering
Eq.(7.7). Since f 00 (t) = − sin tH(t) = −f (t)
L{f 00 (t)}(s) = −L{f }(s) = s2 L{f }(s) − sf (0) − f 0 (0)
| {z }
=1
noindent and so
L{sin tH(t)} =
s2
1
+1
Re{s} > 0
7.3.4
Laplace transform and integration
Laplace Transform
124
Theorem 7.3.4 Let g(t) =
Rt
0
f (t0 )dt0 the primitive of f which vanish at t = 0,
then
L{
Z t
L{f }(s)
s
f (t0 )dt0 }(s) =
0
(7.8)
Proof
We have g 0 (t) = f (t) and g(0) = 0. It follows that
L{g(t)0 }(s) = sL{g}(s) = L{f }(s)
Example 7.10 — Laplace transform of the Fresnel integrals. Consider the Fres-
nel integrals given by
s
s
2 Z t cos τ
√ dτ
C(t) = H(t)
π 0
τ
S(t) = H(t)
2 Z t sin τ
√ dτ
π 0
τ
Z t ±iτ
e
√ dτ
τ
AF
t 7→ f± (t) = H(t)
T
To compute their Laplace transform , we can consider first the function
using 7.8 and 7.7
0
DR
√
1
1
L{f± }(s) = L{H(t)e±it / t}(s) =
s
s
Finally, since C(t) =
q
2 f+ (t)+f− (t)
π
2
and S(t) =
q
s
π
s∓i
2 f+ (t)−f− (t)
,
π
2i
√
1
L{C(t)}(s) =
s
s
1
2π
s
π
+
s−i
s
π
s+i
!
1
=
s
1
L{S(t)}(s) =
is
s
1
2π
s
π
−
s−i
s
π
s+i
!
1
=
is
we obtain
√
s+i+ s−i
√
2s2 + 2
√
√
s+i− s−i
√
2s2 + 2
7.3.5
Laplace transform of a convolution
Theorem 7.3.5 Suppose two causal functions f and g. Then the convolution
R
(f ∗ g)(t) = 0t f (t0 )g(t − t0 )dt0 is causal as well and its Laplace transform is given
by
L{f ∗ g}(s) = L{f }(s)L{g}(s)
Re{s} > α
(7.9)
where α is the largest of the two abcissa of convergence of L{f } and L{g}.
7.4 Aymptotic behaviour
125
Example 7.11 — Laplace transform of tH(t) ∗ t2 H(t). Consider the convolution
g between the causal functions f (t) = tH(t) and h(t) = t2 H(t). It is given by
g(t) =
Z ∞
0
0
0
f (t )h(t − t )dt =
0
Z t
0
t4
t (t − t ) dt = H(t)
12
0
0 2
0
its Laplace transform can be obtained as:
L{
t4
4!
2
H(t)}(s) =
= 5
5
12
12s
s
Re{s} > 0
the same result can be rapidly obtained by using 7.9:
L{
t4
H(t)}(s) = L{tH(t) ∗ t2 H(t)}(s) = L{tH(t)}(s)L{t2 H(t)}(s)
12
1 2!
2
= 5 Re{s} > 0
2
3
s s
s
7.4.1
Aymptotic behaviour
Behaviour at infinity
Let f be a causal function whose Laplace transform L{f } has a minimum abcissa
of convergence α. Thus, for Re{s} > α L{f } is holomorphic and bounded. In
addition, it tends to zero at infinity.
DR
7.4
AF
T
=
Theorem 7.4.1 Let f be a causal function whose Laplace transform’s minimum
abcissa of convergence is α. Then
lim L{f }(s) = 0
|s|→∞
Re{s} > α
(7.10)
Proof
Indeed, let us write s = σ0 + Reiθ with σ0 ∈ R. If |θ| < π2 , then
lim |f (t)e−st | = lim |f (t)|e−σ0 t e−R cos θt = 0
|s|→∞
R→∞
since cos θ > 0. Applying the dominated convergence’s theorem yields the result.
For the case θ = π/2, the Laplace transform can be written as a Fourier transform
of an integrable function and Riemann-Lebesgue’s theorem assures 7.10.
Remark
The function z ∈ C 7→ 1 can’t be the Laplace transform of any function, since
Laplace Transform
126
it does not decay to zero at infinity. We will see, however, that extending the
Laplace transform to distributions, it is possible to show that L{δ} = 1.
7.4.2
The final value theorem
As in the case of the Fourier transform in L1 , there is a correspondance between
the behaviour of a function f at infinity (or at t = 0) and the behaviour of its
Laplace transform in |s| = 0 (or |s| = +∞).
Theorem 7.4.2 Let f be a causal function such that f and f 0 admit a Laplace
transform. If f has a limit f (+∞) when t → +∞, then L{f } verifies
lim sL{f }(s) = f (+∞)
|s|→0+
lim
|s|→0
0
R+
−st
f (t)e
dt =
Z
R+
f 0 (t)dt = f (+∞) − f (0+ )
AF
Z
T
Proof
For Re{s} > 0, |f 0 (t)e−st | ≤ |f 0 (t)|. The dominated convergence theorem then
gives
on the other hand, L{f 0 }(s) = sL{f }(s) − f (0+ ) which gives
lim
|s|→0
7.4.3
R+
0
f (t)e
−st
dt = lim (sL{f }(s) − f (0+ )) = f (+∞) − f (0+ )
DR
Z
|s|→0
The initial value theorem
Theorem 7.4.3 Let f be a causal function such that f and f 0 admit a Laplace
transform. If f has a limit f (0+ ) when t → 0, then L{f } verifies
lim sL{f }(s) = f (0+ )
|s|→+∞
Proof
The Laplace transform of f 0 is L{f 0 (t)}(s) = sL{f }(s) − f (0+ ). Since the Laplace
transform tends to zero when |s| → +∞, we conclude
lim (sL{f }(s) − f (0+ )) = 0
|s|→+∞
Example 7.12 — Asymptotic values of the Heaviside function. For the Heavi-
side function H(t) one has:
H(0+ ) = H(+∞) = 1
Its Laplace transform is L{H}(s) =
1
s
Re{s} > 0 and verifies
7.5 The inverse Laplace transform
127
lim sL{H}(s) = 1 = H(0+ )
|s|→+∞
and
lim sL{H}(s) = 1 = H(+∞)
|s|→0+
Example 7.13 — Asymptotic values of cos t. For the function f (t) = cos(t) one
has:
f (0+ ) = 1
Its Laplace transform is L{cos t}(s) =
s
s2 +1
and indeed
lim sL{cos(t)}(s) = 1 = f (0+ )
|s|→+∞
on the other hand, f (+∞) does not exists, whereas
T
lim sL{cos(t)}(s) = 0
The inverse Laplace transform
Theorem 7.5.1 — The Fourier-Mellin integral. Let f be a causal function with
DR
7.5
AF
|s|→0+
Laplace transform L{f } whose minimum abcissa of convergence is α, and such
that the function sigma : R 7→ G(σ + iω) is integrable over R for every σ > α.
Let Γ be a line parallel to the imaginary axis of abcissa σ > a, called Bromwhich
line. The inverse Laplace transform is then given by the Fourier-Mellin integral.
For almost every value of t, we have
f (t) =
Z σ+iω
1 Z
1
L{f }(s)est ds =
lim
L{f }(s)est ds
2πi Γ
2πi ω→∞ σ−iω
(7.11)
Proof
According to Eq.(), for σ > α we can write
L{f }(σ + iω) =
√
2πF{f (t)H(t)e−σt }(ω)
and since ω : R 7→ L{f }(σ + iω) is integrable, Fourier inversion formula gives
f (t)H(t) =
eσt Z
L{f }(σ + iω)eiωt dω
2π R
and writing s = σ + iω , ds = idω and we obtain
Laplace Transform
128
Z σ+iω
1 Z
1
st
L{f }(s)e ds =
f (t)H(t) =
lim
L{f }(s)est ds
ω→∞
2πi
2πi Γ
σ−iω
1
+
n
Example 7.14 — Inverse of 1/s . Consider G(s) = n with n ∈ N . We see that
s
G is holomorphic everywhere in the complex plane except at s = 0, and tends to
zero when |s| → ∞. For every σ > 0, the function
ω ∈ R 7→
1
1
=
|s|n
(σ 2 + ω 2 )n/2
∈ L1 (R; C)
and the inversion formula for the Laplace transform can be used. We have
1 I est
1 Z est
ds = lim
ds
g(t) =
R→0 2πi C sn
2πi Γ sn
T
where the closed curve C is composed by a vertical segment for which Re{s} = σ0 >
0, completed by a semi-circular path of radius R, and whose contribution to the
total integral vanish when R → ∞. For t < 0, this is true if the semicircular path
is contained in the region Re{s} > σ0 . Since the function est /sn is holomorphic in
that region, Cauchy’s residues theorem gives
I st
e
1
lim
ds = 0 t < 0
2πi R→0 | C s{zn }
AF
g(t) =
=0
DR
For t > 0, however, the contribution of the circular segment to the closed integral
vanish when R → ∞ if the curve is contained in the region Re{s} < σ0 . In this
case, the closed curve C contains a pole of order n at s = 0 of est /sn . Cauchy’s
resiudes theorem gives
I st
1
e
est
g(t) =
lim
ds
,s = 0
=
Res
2πi R→0 | C s{zn }
sn
!
t>0
=0
recalling that the residue of a function at a pole s = a of orther n is
Res (f (s), s = a) =
we obtain
1
∂ n−1
lim n−1 ((s − a)n f (s))
(n − 1)! s→a ∂s
est
tn−1
g(t) = Res n , s = 0 =
t>0
s
(n − 1)!
!
We conclude that the inverse Laplace transform of G(s) =
g(t) =
1
,
sn
n ∈ N+ is
tn−1 H(t)
(n − 1)!
7.6 The Laplace transform of a distribution
129
Example 7.15 — Inverse laplace transform of s/s2 + 4. Consider F (s) =
s
s2 +4
for Re{s} > 0. Its inverse f can be calculated with Fourier-Mellin’s integral
f (t) =
1 Z
z
ezt dz
2
2πi Γ z + 4
with Γ a vertical curve of the form z = σ + iω , ω ∈ (−∞, inf ty) and σ > 0. The
function z ∈ C 7→ zezt /(z 2 + 4) = zezt /((z + 2i)(z − 2i)) has two simple poles at
z = ±2i. The integral may be evaluated by considering the closed paths γ shown
in the figure below, and taking the limit when R → ∞. Cauchy’s residue theorem
gives, for t > 0
f (t) =
1 Z
z
z
z
1 2it
zt
zt
zt
−2it
e
dz
=
Res(
e
,
2i)+Res(
e
,
−2i)
=
e
+
e
2πi Γ z 2 + 4
z2 + 4
z2 + 4
2
whereas for t < 0, no poles are inside γ and f (t) = 0. Finally
The Laplace transform of a distribution
Let us recall that the support supp(T) of a distribution T is the smallest closed
0
ensemble outside which the distribution is zero. We note D+ (R) the set of distributions whose support is included in [0, +∞[.
DR
7.6
AF
T
f (t) = cos 2tH(t)
0
The Laplace transform of a distribution T ∈ D+ (R) with compact support is the
function s ∈ C → T (s) defined by
T (s) = hT, e−st i
(7.12)
Remarks
• In the case of a regular distribution, the definition coincides with the Laplace
transform for functions.
• The function s 7→ e−st is C ∞ (C; C) but has not a compact support. T (s) is
well defined because the distribution T itself has compact support.
Example 7.16 — Laplace transform of a Dirac. Consider the dirac distribution
δ, whose support is supp(δ) = {0} ∈ [0, ∞). Its Laplace transform L{δ}(s) reads
L{δ}(s) = hδ, e−st i = 1
Laplace Transform
130
Laplace transform of a distribution and differentiation
Theorem 7.6.1 Let T be a distribution admitting a Laplace transform. The
derivative D1 T admits a Laplace transform as well that verifies
L{D1 T}(s) = sT (s)
(7.13)
the function s 7→ T (s) is holomorphic in all the complex plane, and one has
T 0 (s) = h−tT, e−st i
(7.14)
the generalization for a nth order derivative reads
L{Dn T}(s) = sn T (s)
dn T (s)
= h(−t)n T, e−st i
dsn
(7.15)
0
AF
T
Proof Indeed, if T is a distribution of compact support belonging to D+ (R), the
same is true for its derivative. D1 T then admits a Laplace transform, and
hD1 T, e−st i = −hT, (e−st )0 i = −hT, −se−st i = shT, e−st i = sT (s)
and, similarly
DR
7.6.1
h−tT, e−st i = hT, −te−st i = hT,
d −st
dT (s)
(e )i =
ds
ds
Applyting this result n times allows to demonstrate (7.15).
Remark
Eq.(7.13) is similar to the property for causal funcions (7.3.3) except that for the
contribution of f (0+ ) which is already taken into account in the framework of
distributions. Indeed, let f = gH be a causal function where g ∈ C 1 (R, C) and H
is the heaviside function. It defines a regular distribution f such that
L{D1 f }(s) = sL{f }(s)
on the other hand, the jump formula 5.18 gives
D1 f = f 0 + f (+0)δ
so that
L{D1 f }(s) = L{f 0 }(s) + f (+0)L{D1 δ}(s) = L{f 0 }(s) + f (+0) = sL{f }(s)
and we find
7.6 The Laplace transform of a distribution
131
L{f 0 }(s) = sL{f }(s) − f (+0)
since f and f 0 are two regular distributions, L{f 0 }(s) = L{f 0 }(s) and L{f }(s) =
L{f }(s) so we recover the usual property for the Laplace transform of functions
Example 7.17 — Laplace transform of Dn δ. Using 7.13 we obtain, for n = 1
L{D1 δ} = sL{δ} = s
and in the general case
L{Dn δ} = sL{Dn−1 δ} = sn
AF
T
In principle, such a distribution is a linear continuous form over the space
of test functions, which have themselves compact support. However, since the
distribution itself has compact support, one can ignore the test fonctions outside
the support of the distribution. This allows to extend the duality and consider the
distributions with compact support as the linear and continuous forms over the
space of functions C ∞ (R; C)
hT, ϕi := hT, χϕi
DR
where χ ∈ D(R; C) such that χ = 1 in an open vicinity of supp(T). This definition
is independent on the choice of χ.
T
AF
DR
DR
AF
T
8 — Appendix
Appendix
134
8.1
Vector spaces
A vector space V over a Field K (K = R or K = C in this book) satisfy the
following axioms for the addition and scalar multiplication:
• Associativity of addition
f + (g + h) = (f + g) + h
• Commutativity of addition
f +g =g+f
∀f, g, h ∈ V
∀f, g ∈ V
• Identity element of addition
∀f ∈ V
there exists 0 ∈ V such that f + 0 = f
• Inverse elements of addition
that f + (−f ) = 0
For every f ∈ V , there exists −f ∈ V such
• Compatibility of scalar and field multiplication
V, λ, η ∈ K
1f = f for all f ∈ V , where 1
T
• Identity element of scalar multiplication
is the multiplicative identity in F
∀f ∈
λ(ηf ) = (λη)f
AF
• Distributivity of scalar multiplication with respect to vector addition
λ(f + g) = λf + λg ∀f, g ∈ V, λ ∈ K
8.2
DR
• Distributivity of scalar multiplication with respect to field addition
(λ + η)f = λf + ηf
∀f ∈ V, λ, η ∈ K
Convolution between functions
The convolution between two functions f and g is defined by
f ∗ g(t) =
Z
f (t − u)g(u)du
R
provided that the integral exists. This integral is absolutely convergent (
u)g(u)|du < ∞) if:
R
R
|f (t −
• If f ∈ L1 (R; C) and g is bounded.
• If f and g are both in ∈ L2 (R; C)
• The product h = f g of an integrable function f ∈ L1 (R; C) with a continous
and bounded function g (g ∈ Cb (R; C)) belongs to L1 (R; C).
• If f and g are both in ∈ L1 (R; C), then f ∗g is well defined almost everywhere
and belongs to L1 (R; C).
8.3 Convolution between distributions and functions
135
• The support of a convolution is bounded: Let f and g be two mesurable
functions defined for almost every t over R for which the convolution f ∗ g is
well defined. Then
supp(f ∗ g) ⊂ supp(f ) + supp(g)
(8.1)
If one of the two functions has compact support, then
supp(f ∗ g) ⊂ supp(f ) + supp(g)
(8.2)
• For function ϕ ∈ D(R; C) and f inL1loc (R; C), the convolution ϕ ∗ f belongs
to C ∞ (R) and
dα
dα
(ϕ
∗
f
)(t)
=
(
f ) ∗ f (t)
dtα
dtα
!
∀α ∈ N
also
T
ϕ ∈ Ccm (R; C) and f inL1loc (R; C) → ϕ ∗ f ∈ C m (R)
(8.3)
AF
• Associativity of the convolution: Let f, g ∈ D(R; C) and h inL1loc (R; C).
Then
f ∗ (g ∗ h) = (f ∗ g) ∗ h
(8.4)
DR
Indeed, the term g ∗ h ∈ C ∞ (R), which is then a locally integrable function,
and so f ∗ (g ∗ h) is well defined since f ∈ D(R). On the other hand, f ∗ g is
a continuous function with compact support according to 8.2 and 8.3 , and
so (f ∗ g) ∗ h is well defined since h inL1loc (R; C).
8.3
Convolution between distributions and functions
• For every distribution T ∈ D0 (R) and every test function ϕ ∈ D(R; C), we
define
(T ∗ ϕ)(x) = hT, ϕ(x − ·)i
∀x ∈ R
(8.5)
where ϕ(x − ·)i = τ−x ∂−1 ϕ. In addition
supp(T ∗ ϕ) ⊂ supp(T) + supp(ϕ)
(8.6)
• Regularity of the convolution: For every distribution T ∈ D0 (R) and every
test function ϕ ∈ D(R; C), the convolution between T and ϕ is of class
C ∞ (R; C) and
dn
dn
n
(T
∗
ϕ)
=
(D
T)
∗
ϕ
=
T
∗
ϕ
dtn
dtn
(8.7)
Appendix
136
• For every distribution T ∈ E 0 (R) with compact support and every test
function ϕ ∈ C ∞ (R; C), we define
(T ∗ ϕ)(x) = hT, ϕ(x − ·)i
∀x ∈ R
(8.8)
one has T ∗ ϕ ∈ C ∞ (R; C) and
dn
dn
n
(T
∗
ϕ)
=
(D
T)
∗
ϕ
=
T
∗
ϕ
dtn
dtn
• For every distribution T ∈ E 0 (R) with compact support and every Schwartz
function ϕ ∈ S(R; C)
T ∗ ϕ ∈ S(R; C)
(8.9)
with
(T ∗ ϕ)(x) = hT, ϕ(x − ·)i
(8.10)
Convolution between distributions
T
8.4
∀x ∈ R
AF
• For every T ∈ D0 (R) and every S ∈ E 0 (R), we define the convolution product
T ∗ S ∈ D0 (R) by
hT ∗ S, ϕi = hT, (∂−1 S) ∗ ϕi = hS, (∂−1 T) ∗ ϕi
∀ϕ ∈ D(R, C) (8.11)
DR
so that T ∗ S = S ∗ T, and
supp(T ∗ S) ⊂ supp(T) + supp(S)
(8.12)
• For every T ∈ D0 (R) , S ∈ E 0 (R) and α ∈ N
Dα (T ∗ S) = (Dα T) ∗ S = T ∗ (Dα S)
(8.13)
• Associativity: If T, S and R are three distributions over R, and if at least
two of them have compact support, then
R ∗ (S ∗ T) = (R ∗ S) ∗ T
8.5
Monotone convergence theorem
Theorem 8.5.1 — Bleppo Levi theorem for monotone convergence. Let {fn }n≥1
be a non-decreasing sequence of positive functions fn : R 7→ R≥0 , that is, for
every t ∈ R and n ≥ 1
0 ≤ fn (t) ≤ fn+1 (t) ≤ ∞
then
8.6 Differentiation under the integral sign
Z
137
lim fn (t)dt = n→∞
lim fn (t)dt ≤ +∞
R n→∞
The theorem remains true if the assumptions hold for almost every value of t.
Corollary 8.5.2 If uj : R 7→ R≥0 are positive functions, then
Z X
∞
uj (t)dt =
R j=1
8.6
∞ Z
X
uj (t)dt ≤ +∞
j=1 R
Differentiation under the integral sign
Theorem 8.6.1 — Differentiation under the integral sign. Let I and A be an
interval and a subset of R, respectively. Let f be a function defined over A × I
vérifying the following hypothesis:
AF
T
(a) For every λ ∈ I, the function t 7→ f (t, λ) is integrable over A
(b) The partial derivative ∂f /∂λ(t, λ) exists at every point of A × λ
(c) There is a positive and integrable function h over A such that |∂f /∂λ(t, λ)| ≤
h(t) for any λ and t.
Then the function F defined by
DR
F (λ) =
Z
f (t, λ)dt
A
is differentiable in I, and one has
0
F (λ) =
Z
A
8.7
∂f
(t, λ)dt
∂λ
Dominated convergence theorem
Theorem 8.7.1 — Dominated convergence theorem. Let (fn )n∈N be a sequence
of integrable functions that converges pointwise to a function f . If there is an
integrable function g such that
|fn (t)| ≤ g(t)
∀n ∈ N, ∀t ∈ R
then
lim
Z
n→∞ R
which also implies
|fn − f (t)|dt = lim kf − fn kL1 = 0
n→∞
Appendix
138
lim
n→∞
8.8
Z
fn (t)dt =
Z
R
f (t)dt
R
Fubini theorem
Theorem 8.8.1 — Fubini’s theorem. Let f be a function defined over R × R. If
f is integrable over R2 , then:
ZZ
R2
8.9
f (t, u)dtdt =
Z Z
R
f (t, u)dt du =
R
Z Z
R
f (t, u)du dt
(8.14)
R
Approximation of the identity
Theorem 8.9.1 — Approximation of the identity. Let be h ∈ L1 (R; C) a function
such that
R
R
h(t)dt = 1. For > 0, let’s define
T
1
h = h(t/)
The Riemann-Lebesgue theorem
DR
8.10
AF
then, for every function f ∈ L1 (R; C) (L1 (R; C)), the sequence of functions f ∗ h
converge to f in L1 (R; C) (L1 (R; C)) when tends to zero. The sequence f ∗ h
is called approximation of the identity.
Theorem 8.10.1 — Riemann-Lebesgue. For every f ∈ L1 (R; C), its Fourier
transform fˆ verifies:
a) fˆ ∈ Cb (R; C) (the space of continuous and bounded functions), indeed fˆ is
bounded by the L1 -norm of f , |fˆ(ω)| ≤ √12π kf kL1 for every ω ∈ R.
b) fˆ(ω) → 0 when |ω| → 0
8.10.1
Proof of the Riemann-Lebesgue theorem
a) Let (ωn )n∈N be a sequence in R such that limn→∞ ωn = ω. Then, for almost
every t ∈ R, we have:
1
1
lim √ f (t)e−iωn t = √ f (t)e−iωt
2π
2π
n→∞
and
|f (t)e−iωn t | ≤ |f (t)|
since f ∈ L1 (R; C), the dominated convergence theorem (see 8.7.1) gives
8.10 The Riemann-Lebesgue theorem
139
1 Z
1 Z
−iωn t
√
√
lim
f (t)e
=
lim f (t)e−iωn t = fˆ(ω)
n→∞
n→∞
2π R
2π R
so that
lim fˆ(ωn ) = fˆ(ω)
n→∞
since the sequence (ωn )n∈N → ω is arbitrary, one concludes that fˆ is continous
over R. Moreover, for every ω ∈ R:
1 Z
1
ˆ
√
|f (t)|dt = √ kf kL1
|f (ω)| ≤
2π R
2π
so that fˆ is bounded by kf kL1 . Since fˆ is therefore always a bounded function, it
belongs to L∞ (R; C) and its L∞ -norm verifies:
T
1
kfˆk∞ = sup fˆ(ω) ≤ √ kf kL1
2π
ω∈R
AF
b) For ω 6= 0 we have:
1 Z
1 Z
−iω(t+π ω 2 )
−iωt
ˆ
|ω|
f (t)e
dt = − √
f (t)e
dt
f (ω) = √
2π R
2π R
2 /|ω|2 π
= e−iπ = −1. Changing variables with y = t +
DR
since e−iω
πω
|ω|2
1 Z
πω −iωy
1 Z
πω −iωt
√
fˆ(ω) = − √
f (y −
)e
dy
=
−
f (t −
)e
dt
2
|ω|
|ω|2
2π R
2π R
so that:
1 Z
πω
ˆ
√
f (ω) =
(f (t) − f (t −
))e−iωt dt
2
|ω|
2 2π R
1 Z
πω
1
|fˆ(ω)| ≤ √
|f (t) − f (t −
)|dt ≤ √ kf − τπω/|ω|2 f kL1
2
|ω|
2 2π R
2 2π
where τ is the translation operator defined in 1.12. Clearly
lim kf − τπω/|ω|2 f kL1 = 0
|ω|→∞
so that
lim |fˆ(ω)| = 0
|ω|→∞
Appendix
140
Proof of Fourier’s inversion formula in L1
2
Let G be the Gaussian function G(ω) = √12π e−ω /2 and consider the sequence
(ωn )n∈N = ω/n ∈ R such that limn→∞ ωn = 0. Then, for every t ∈ R:
1
lim eiωt fˆ(ω)G(ωn ) = eiωt fˆ(ω)G(0) = √ fˆ(ω)
2π
n→∞
and
|fˆ(ω)eiωt G(ωn )| ≤ |fˆ(ω)G(ω)|
∀n ∈ N
and since fˆG belongs to L1 (R; C),the dominated convergence theorem (see 8.7.1)
gives
1 Z ˆ
lim
fˆ(ω)eiωt G(ω/n)dω = √
f (ω)eiωt dω = f (−t)
n→∞ R
2π R
V
V
Now
Z
(
Z
e
iωt
)
1 Z −iωu
√
e
f (u)du G(ω/n)dω
2π R
AF
fˆ(ω)eiωt G(ω/n)dω =
T
Z
R
R
Using Fubini’s theorem:
Z
1 Z
iωt
i(t−u)ω
ˆ
f (ω)e G(ω/n)dω = √
e
G(ω/n)dω f (u)du
R
2π R R
Z
DR
8.11
and making the substitution ω/n = x
Z
1 Z
fˆ(ω)eiωt G(ω/n)dω = √
n
ei(t−u)nx G(x)dx f (u)du
R
R
2π R
Z
one recognizes the Fourier transform of G at ω = (t − u)n. Since the Fourier
transform of a Gaussian is also a Gaussian (see example 2.6), we get
Z
fˆ(ω)eiωt G(ω/n)dω =
R
Z
nG(n(t − u))f (u)du = (Gn ∗ f )(t)
R
where Gn (t) = nG(nt), which is positive and integrable of integral equal to 1.
(Gn )n∈N is an approximation of the identity (see 8.9.1), therefore
lim kGn ∗ f − f kL1 = 0
n→∞
so that there exists a sub-sequence nk such that for almost every value of t
V
V
lim (Gnk ∗ f )(t) = f (t) = f (−t)
k→∞
8.12 Complex analysis
141
V
V
and f (−t) = f (t), which demonstrates the theorem.
8.12
Complex analysis
8.12.1
Derivative of a complex-valued function
Suppose complex-valued function f : C → C. The derivative of f at point z = z0
is given by
lim
z→z0
f (z − z0 )
z − z0
if the limit exists, the function f is said to be complex-differentiable at point
z = z0 .
Holomorphic and analytic functions
T
8.12.2
AF
• A complex-valued function f : C → C is holomorphic over the open set
U ⊆ C if it is complex-differentiable at everypoint z0 of U .
• A remarkable result due to Cauchy is that a function is holomorphic over
an open set U ⊆ C if and only if is also analytic, that is, an infinitely
differentiable function that is locally given by a convergent power series:
DR
n
∞
X
( d f n(z) )z=z0
f (z) =
dz
n=0
n!
(z − z0 )n
∀z0 ∈ U
Note that the equivalence between holomorphic and analytic is a property
of complex-valued functions which does not hold in general for real-valued
function.
• An entire function is a complex-valued function that is holomorphic over the
entire complex plane. This is the case, for example, of any complex polynomial, the complex exponential, the complex trigonometric or hyperbolic
functions.
P (z)
• Every rational function of the form f (z) = Q(z)
(where P and Q 6= 0 are
two polynomial functions) is holomorphic over the complementary of the set
of all its roots of its denominator, called poles. For example, the function
f (z) = 1/z is holomorphic on all C except at the pole z = 0.
8.12.3
Cauchy-Goursat’s integral theorem
Let U be a simply connected open subset of C , f : U 7→ C a continuous,complexdifferentiable function 1 , and γ a closed rectifiable2 path in U . Then
1
the theorem also holds if f is differentiable except in a finite number of points
A curve γ is rectifiable if there is a smallest number L that is an upper bound on the length
of any polygonal approximation of γ. The number L is defined as the arc length
2
Appendix
142
H
γ
f (z)dz = 0
(8.15)
this leads to the Cauchy’s integral formula and the residue theorem stated below.
Cauchy’s integral formula
Cauchy’s integral formula shows that the value at any point z0 of an holomorphic
function is completely determined by the values the latter takes on a closed path
containing the point z0 . It also allows to write the derivatives of an holomorphic
function in terms of an integral.
Theorem 8.12.1 — Cauchy’s integral formula. Let U be a simply connected
H f (z)
1
dz
2πiI(γ,a) γ z−a
(8.16)
AF
f (a) =
T
open subset of C , f : U 7→ C an holoporhic function, γ a closed path in U and
a a point in U which does not belong to γ. Then
where I(γ, a) is the winding number of γ around a, an integer that represents
the total number of times that the curve γ travels counterclockwise around the
point a (it changes sign if the curve travels around a clockwise).
Remarks
DR
8.12.4
• Cauchy’s integral formula can be used to prove that any holomorphic function
over an open subset U of C is analytic over U . In particular, f is infinitely
differentiable and
dn f
dz n
!
=
z=a
n! I
f (z)
dz
2πi γ (z − a)n+1
where γ is any closed, rectifiable curve in U with winding number 1 around
a.
• To proof of Cauchy’s integral formula one defines the function g as g(z) =
(f (z)−f (a))/(z −a) for z =
6 a and f 0 (a) if z = a. The function g is continuous
over U and holomorphic over U {a}, and we may apply Cauchy-Goursat’s
integral theorem:
I
γ
g(z)dz = 0 =
I
γ
I
f (z)
1
dz − f (a)
dz
z−a
γ z −a
|
{z
2πiI(γ,a)
}
• This formula allows to demonstrate the residue theorem, stated below.
8.13 Proof of the Dirichlet-Jordan’s theorem
8.12.5
143
Pole of an holomorphic function
Let U an open subset of C, a an element of U , and f : U {a} 7→ C an holomorphic
function. The point a is a pole of f if f is not bounded around a (|f (z)| tends to
infinity when z → a) while 1/f remains bounded around a. More precisely, we say
that f admits a pole in a if there exists an holomorphic function g over U such
that g(a) 6= 0 and a non-zero integer n such that for every z ∈ U {a}
f (z) =
g(z)
(z − a)n
the integer n is called order of the pole.
8.12.6
Residue of a function around a pole
If f has a pole of order n in z = a, then the residue Res(f, a) of f at the point
z = a can be calculated as follows
1
dn−1
lim n−1 ((z − a)n f (z))
(n − 1)! z→a dz
T
Res(f, a) =
(8.17)
AF
In particular, if f has a pole of order 1 in z − a, then
8.12.7
DR
Res(f, a) = z→a
lim (z − a)f (z)
Cauchy’s residue theorem
Let U be a simply-connected, open subset of the complex plan C ,{z1 , z2 , ..., zn } a
finite set of points of C and F an holomorphic function over U \ {z1 , z2 , ..., zn } .
If γ is a closed rectifiable curve in U that does not encounter any of the singular
points zk , then
I
F (z)dz = 2πi
γ
n
X
Res(F, zk )I(γ, zk )
(8.18)
k=1
where Res(F, zk ) is the residue of F in zk and I(γ, zk ) is the winding number of γ
around zk .
8.13
Proof of the Dirichlet-Jordan’s theorem
The Dirichlet kernel is defined as:
DN (t) =
N
1 X
sin((N + 1/2)t)
einω0 t =
T n=−N
T sin(t/2)
(8.19)
Appendix
144
with ω0 = 2π/T . The convolution of Dn with a T −periodic function f (T = 2π/ω0 )
is such that:
N
X
1 Z T inω0 (t−y)
1ZT
f (y)DN (t − y)dy =
e
f (y)dy
(DN ∗ f )(t) =
T 0
n=−N T 0
(DN ∗ f )(t) =
N
X
inω0 t
e
n=−N
N
X
1 Z T −inω0 y
e
f (y)dy =
cn (f )einω0 t
T 0
n=−N
so that DN ∗ f (t) is the N th -degree Fourier series approximation of f
(DN ∗ f )(t) =
N
X
cn (f )einω0 t = SN (f )(t)
n=−N
Now, this can be rewritten once again as
AF
T
f (t − u)
1 Z T /2
sin((N + 1/2)t)
du
SN (f )(t) =
T −T /2
sin(t/2)
DR
f (t−u)
Note that the function t 7→ sin(t/2)
is not necessarily integrable around t = 0. Let
us split the integral in two parts:
f (t − u)
f (t − u)
1 Z T /2
1Z0
sin((N +1/2)t)
sin((N +1/2)t)
SN (f )(t) =
du+
du
T 0
sin(t/2)
T −T /2
sin(t/2)
and making a change of variables one obtains:
SN (f )(t) =
1 Z T /2
[f (t − u) + f (t + u)]
sin((N + 1/2)t)
du
T 0
sin(t/2)
Let us define ϕ(t) = 12 (f (t + 0) + f (t − 0)) which is well-defined since f is supposed
to have bounded variation over [0, T ]. One has:
SN (f )(t)−ϕ(t) =
[f (t − u) + f (t + u)]
1 Z T /2
(f (t + 0) + f (t − 0))
sin((N +1/2)t)
du−
T 0
sin(t/2)
2
1 Z T /2
[f (t − u) + f (t + u)] − f (t + 0) − f (t − 0)
SN (f )(t)−ϕ(t) =
sin((N +1/2)t)
du
T 0
sin(t/2)
(t+0)−f (t−0)
and since t 7→ [f (t−u)+f (t+u]−f
is now integrable, the Riemann-Lebesgue
sin(t/2)
theorem assures that:
8.13 Proof of the Dirichlet-Jordan’s theorem
lim SN (f )(t) − ϕ(t) = 0
N →∞
finally, for every t:
N
X
1
cn (f )einω0 t = (f (t + 0) + f (t − 0))
N →∞
2
n=−N
DR
AF
T
lim
145
T
AF
DR