Appel W. Mathematics for Physics and Physicists

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Chapter

Fourier transform

of functions

This chapter introduces an integral transform called the “Fourier transform,” which

generalizes the Fourier analysis of periodic functions to the case of f u nct ions deﬁned

on the whole real axis R.

We start with the deﬁnition of the Fourier transform for functions which are

Lebesgue-integrable (elements of L

). One problem of the Fourier transform thus

deﬁned is that it does not leave t h e space L

stable. It will then be e xtended

to functions which are square integrable (elements of L

), w h ich have a physical

interpretation in terms of energy. The space L

being stable, any square integrable

function has a Fourier transform which is also square integrable.

10.1

Fourier transform of a function in L

In this section, we start by deﬁ ning the Fourier transform of a n integrable

function and deriving its main properties.

Although the Fourier transform is a generalization of the notion of a

Fourier series , it is not the L

or Hilbert space setting which is the simplest.

Therefore we start with functions in L

(R),

before generalizing to square

integrable functions.

The reason is that, in contrast with functions deﬁned on a ﬁnite interval, L

(R) is not

contained in L

(R).

278 Fourier transform of functions

10.1.a Deﬁniti on

DEFINITION 10.1 Let f be a f unction, real- or complex-valued, depending on

one real variable. The Fourier transform (or spectrum) of f , if it exists, is the

complex-valued function deﬁned for the real variable ν by

f (ν)

def

∞

−∞

f (x) e

−2πiν x

dx for all ν ∈ R. (10.1)

This is denoted symbolically

f = F [ f ] or

f (ν) = F



f (x)



Similarly, the conjugate Fourier transform of a function f is given (when it

exists) by

F [ f ] (ν)

def

∞

−∞

f (x) e

2πiν x

dx.

Remark 10.2 The integral that ap pears does not always exist. For instance, if we consider

f : x 7→ x

, the integral

+∞

−∞

−2πiν x

does not exist, for any value of ν, and f does not have a Fourier transform (in the sense of

functions; i t will be seen in the next chapter how to deﬁne i ts Fourier transform in the sense

of distributions).

It is also possible to have to deal with non-integrable functions for which the i ntegral

deﬁning

f (ν) converges as an improper integral. As an example, consider g : x 7→ (sin x)/x.

Then g is not integrable but, on the other hand, the limit

lim

M→+∞

−M

sin x

−2πiν x

exists for all values of ν. More details about this (not a complete description of what can

happen! this is very difﬁcult to obtain and state) will be gi ven in Section 10.3, which discusses

the extension of the Fourier transform to the space of square integrable functions.

At least if f is integrable on R, it is clear that

f is deﬁned on R. Indeed, recall that for a

function f : R → R, the following equivalence holds:

f is Lebesgue-integrable ⇐⇒ |f | is Lebesgue-integrable,

which shows that if f is integrable, then so is x 7→ f (x) e

−2πiν x

for any real value of ν.

THEOREM 10.3 (Fourier transform of an integrable function) Let f be a func-

tion which is Lebesgue-integra ble on R. Then the Fourier transform ν 7→

f (ν) of f is

deﬁned for all ν ∈ R.

Remark 10.4 A number of different conventions regarding the Fourier transform are in use in

various ﬁelds of physi cs and mathematics; in the table on page 612, the most common ones

are presented with the corresponding formula for the inverse Fourier transform.

Fourier transform of a function in L

279

10.1.b Examples

Example 10.5 Consider the “rectangle” function (already considered in Chapter 7):

Π(x)

def

(

0 for |x| >

1 for |x| <

Then its Fourier transform is a sine cardinal (or cardinal sine):

Π(ν) =

−

−2πiν t

dt = sinc(πν)

def







sin πν

πν

if ν 6= 0,

1 if ν = 0.

Similarly, the Fourier transform of the characteristic function χ

[a,b ]

of an interval is

eχ

[a,b ]

(ν) = F



[a,b ]



(ν) =







sin



π(b −a)ν



πν

−iπ(a+b)ν

if ν 6= 0,

b −a if ν = 0.

Example 10.6 Consider the gaussian x 7→ e

−πx

. Then one can show (see Exercice 10.1 o n

page 295 or, usi ng the method of residues, Exercise 4.13 on page 126) that



−πx



= e

−πν

Example 10.7 The lorentzian function with parameter a > 0 is deﬁned by

f (x) =

+ 4π

Then, by the method of residues, one can show that



+ 4π



= e

−a|ν|

The table on page 614 lists some of the most useful Fourier transforms.

10.1.c The L

space

Let L

(R) denote temporarily the s pace of functions on R which are Lebesgue-

integrable. Can it be normed by putting

kf k

def



f (t)



dt ?

The answer is “no.” The reason is similar to what happened in the d eﬁnition

of the space L

in the prev ious cha pter: if f and g are two functions in L

which are equal almost everywhere,

then kf − gk = 0, although f 6= g;

this means that the map f 7→ kf k is not a norm. To resolve this d ifﬁculty,

For instance, one can take f = χ

, the characteristic function of the set Q, also called the

Dirichlet function, and g the zero function.

280 Fourier transform of functions

those functions which are equal almost everywhere are identiﬁed. The set of

all functions which are equal to f almost everywhere is the equivalence class

of f. The set of equivalence classes of integrable functions is denoted L

(R)

or more simply L

. Thus, when speaking of a “function” f ∈ L

, what is

meant is in fact any function in the same equivalence class. For instance, the

zero function and the D irichlet function belong to the same equivalence class,

denoted 0. In order to not complicate matters, there are both said to be “ equal

to the zero function.”

In the space L

, the map f 7→

|f | is indeed a norm.

DEFINITION 10.8 The space L

(RR

RR) or L

is the vector space of integrable

functions, deﬁned “up to equality a lmost everywhere,” with th e norm

kf k

def



f (t)



dt.

Consider two functions f and g which coincide a lmost everywhere. Then

f =

g. Moreover, if ϕ is any other function, it follows that f ϕ is equal to

gϕ almost everywhere. So applying this result to the case of ϕ : x 7→ e

−2πiν x

it follows that two functions which are equal almost everywhere ha ve the same Fourier

transform.

Hence, we can consider that the Fourier transform is deﬁned on the

space L

(R) of integrable functions deﬁ ned up to equality almost everywhere.

10.1.d Elementary properties

THEOREM 10.9 (Properties of

f ) Let f ∈ L

(R) be an integrable function and

denote by

f its Fourier transform. Then

f is a continuous function on R. Moreover,

it is bounded, and in fact k

f k

∞

¶ kf k

DEFINITION 10.10 The space L

∞

(RR

RR) is the vector space of bounded measur-

able functions on R, with t he sup norm deﬁned by

kf k

∞

def

= sup

x∈R



f (x)



(It is possible to deﬁne functions of L

∞

up to equality almost everywhere. Th e

deﬁnition of the norm must then b e modiﬁed accordingly. This reﬁnement

is not essential here).

THEOREM 10.11 (Continuity of the Fourier transform in L

) The Fourier trans-

form, deﬁned on L

(R) and taking values in L

∞

(R), is a continuous linear oper-

ator, that is

i) (linearity) for any f, g ∈ L

(R) and any λ, µ ∈ C, we have

F [λ f + µg] = λF [ f ] + µF [g] ;

Fourier transform of a function in L

281

ii) (continuity) if a sequence ( f

)

n∈N

of integrable functions converges to 0 in the

sense of the L

norm, then the sequence (

)

n∈N

tends to 0 in the sense of the

sup norm:

Z



(x)



dx −−−→

n→∞



=⇒



sup

ν∈R



(ν)



−−−→

n→∞



Proof of the two prec eding theorems. To show the continuity of the function

f ,

we apply the theorem of continuity of an integral depending on a parameter (Theo-

rem 3.11 on page 77), using the domination relation

∀x ∈ R



f (x) e

−2πiν x



f (x)



Moreover, for any ν ∈ R, we have



f (ν)



f (x) e

−2πiν x



f (x)



dx = kf k

which shows that

f k

∞

= sup

ν∈R



f (ν)



¶ kf k

Since t he Fourier transform is obviously linear, this inequality is sufﬁcient to establish

its continuity (see Theorem A.51 on page 583 concerning continuity of a linear map).

The continuity of the Fourier transform is an important fact. It means

that if a sequence ( f

)

n∈N

of integrable functions converges to f ∈ L

, in the

sense of the L

norm, namely,



(t) − f (t)



dt −−−→

n→∞

then there will b e convergence in the sense of the L

∞

norm, that is, uniform

convergence of the sequence (

)

n∈N



−→ f



=⇒



cv.u.

−−→

f on R



Example 10.5 on page 279 shows that the Fourier transform of a function

in L

is not necessarily integrable it self. So the “Fourier transform” operator

does not preserve the space L

, which is somewhat inconvenient, as one often

prefers an honest endomorphism of a vector space to an arbitrary linear map.

(It will be seen in Section 10.3 th at it is possible to extend the Fourier trans-

form to the space L

and that L

is then stable by t he Fourier transform th us

deﬁned.)

On the other hand, one can see that each of th e Fourier transforms previ-

ously computed has the property that it tends to 0 at inﬁnity. This property

is conﬁrmed in general by the following theorem, which we will not prove

(see, e.g., [76]). It is the analogue of the Riemann-Lebes gue Lemma 9.44 for

Fourier series.

282 Fourier transform of functions

THEOREM 10.12 (Riemann-Lebesgue lemma) Let f ∈ L

(R) be an integrable

function and denote by

f its Fourier transform. Then the continuous function

f tends

to 0 at inﬁnity :

lim

ν→±∞



f (ν)



= 0.

10.1.e Inversion

The question we now ask is: can one inverse the Fourier transform? The

answer is partially “yes”; moreover, the inverse t ransform is none other than

the conjugate Fourier transform.

We just saw that, even if f ∈ L

(R), its Fourier transform

f is not itself

always integrable, and so the conjugate Fourier transform of

f may not be

deﬁned. However, in the special case where

f ∈ L

, we have the following

fundamental result.

THEOREM 10.13 (Inversion in L

(RR

RR)) Let f ∈ L

(R) be an integrable function

such that the Fourier transform

f of f is itself integrable. Then we have

F [

f ] (x) = f (x) for almost all x;

and also

F [

f ] (x) = f (x) at any point x where f is continuous.

In particular, if f is also continuous,

we have F [

f ] = f .

Proof. We prove the inversion formula only at points of continuity. The proof starts

with the following lemma.

LEMMA 10.14 Let f and g be two integrable functions. Then

f · g and f · eg are integrable

and we have

f (t) eg(t) dt =

f (t) g(t) dt.

Proof of the lemma. Since

f is bounded and g is integrable, the product

f · g is also integrable and similarly for f · eg. Hence by Fubini’s theorem, we

have

f (t) eg(t) dt =

f (t)

Z

g(s) e

−2πi ts



g(s)

Z

f (t) e

−2πi st



ds =

g(s)

f (s) ds,

which proves the lemma.

Then we are not looking at it simply up to “equality almost everywhere.”

Fourier transform of a function in L

283

Now introduce the sequence of functio ns (h

)

n∈N

deﬁned by

(x) = exp(−π

It is a sequence of gaussians which are more and more “spread out,” and converges

pointwise to the constant function 1. Since

f is integrable and



f (x) h

(x) e

2πi xt



f (x)



for any reals x, t ∈ R and any n, Lebesgue’s dominated convergence theorem p roves

that

f (x) h

(x) e

2πi xt

dx −−→

n→∞

f (x) e

2πi xt

dx = F





(t) (10.2)

for any t ∈ R. The Fourier transform of x 7→ h

(x) e

2πi xt

is, as shown by an immediate

calculation, equal to x 7→

(x − t), and the previous lemma yields

f (x) h

(x) e

2πi xt

dx =

f (x)

(x − t) dx. (10.3)

But, as seen in Example 8.17 on p age 231, the sequence of Fourier transforms of h

: x 7−→

−n

which is a Dirac sequence. It follows that, if t is a point of continuity for f , we have

f (x)

(x − t) dx −−→

n→∞

f (t). (10.4)

Putting equations (10.2), (10.3), and (10.4) together, we obtain, for any point where f is

continuous, that

f (x) e

2πi xt

dx = F





(t) = f (t),

which is the second result stated.

COROLLARY 10.1 5 If f is integrable and is not equal almost everywhere to a con-

tinuous function, then its Fourier transform is not integrable.

Proof. If the Fourier transform

f is integrable, then f coincides al most everywhere

with the inverse Fourier transform of

f , which is continuous.

Remark 10.16 With the conventions we have chosen for the Fourier transform, we have, when

this makes sense, F

−1

= F . One could use indifferently F

−1

or F . The reader should be

aware th at, in other ﬁelds, the convention use d for the deﬁnition of the Fourier transform is

different from the one we use (for instance, in quantum mechanics, in optics, or when dealing

with functions of more than one variable), and the notation is not equivalent then. A table

summarizes the main conventions used and the corresponding inversion formulas (page 612).

Remark 10.17 Theorem 10.13 may be proved more simply using Lemma 10.14 and the fact

that the Fourier transform of the constant function 1 is the Dirac δ. But this property will

only be established in the next chapter. For this reason we use instead a sequence of functions

)

n∈N

converging to 1, while the sequence of their Fourier transforms converges to δ.

To apply the inversion formula of Theorem 10.13, one needs some infor-

mation concerning not only f , but also

f (which must be integrable); this may

be inconvenient. In the next proposition, information relative to f sufﬁces

to ob tain the inversion formula.

284 Fourier transform of functions

PROPOSITION 10.18 Let f be a function of C

class such that f , f

′

, and f

′′

are

all integrable. Th en

f is also integrable; the Fourier inversion formula holds, and we

have

f = F [

f ] .

Example 10.19 Consider again the example of the lorentzian function. We have

f (x) =

1 + 4π

Then

′

(x) =

−16 π

(1 + 4π

)

and f

′′

(x) =

192 π

−16 π

(1 + 4π

)

which are both integrable. We deduce from this that F



−|ν|



= 2/(1 + 4π

). Moreover,

since all t h e functions considered are real-valued, it follows by taking the complex conjugate

and exchanging the variables x and ν that also



−|x|



1 + 4π

Hence the inversion formula is also a tool that can be used to compute

new Fourier transforms easily!

10.1.f Exten sion of the inversion formula

We can now discuss the case where f has a Fourier transform

f which is not

integrable, that is, such that

f (ν) e

2πiν x

is not deﬁned in the sense of Lebesgue. In s ome situations, the limit

lim

R→+∞

−R

f (ν) e

2πiν x

dν

may still be deﬁned. (It is not quite an improper integral, because the bounds

of integration are symmetrical.) For example, the Fourier transform of the

“rectang le” function is a cardinal sine. The latter is not Lebesgue-integrable,

but the integral

I(x, R)

def

−R

sin πν

πν

2πiν x

dν

does have a limit as [R → ∞]. In such cas es, the following theorem may be

used.

Properties of the Fourier transform 285

THEOREM 10.20 Let f (x) be an integrable function of class C

except at ﬁnitely

many points of discontinuity, such that f

′

∈ L

. Then for all x we have

lim

R→+∞

−R

f (ν) e

2πiν x

dν =

f (x

) + f (x

−

)

By abuse of notation, this limit will still be denoted F





(x).

In other words, the inversion formula, in the sense of a limit of integrals

(called the principal value), gives the regularized function associated to f .

This should be compared to Dirichlet’s theorem on page 268.

In particular, if f is continuous and f and f

′

are integrable, the inversion

formula of Theorem 10.20 gives back the original function f .

Example 10.21 We saw that the Fourier transform of the “rectangle” function is si nc(πν) .

Since the “rectangle” funct ion has only two discontinuities and since its derivative (in the

sense of functions) is always zero and therefore integrable, it follows that

lim

R→+∞

−R

sinc(πν) e

2πiν x

dν = F



sinc(πν)



(x) =







0 i f |x| >

1 i f |x| <

if |x| =

10.2

Properties of the Fourier transform

10.2.a Transpose and translates

THEOREM 10.22 Let f be an integrable function and a ∈ R a real number. We

have



f (x)



f (−ν), F



f (−x)



= F



f (x)



f (−ν),



f (x −a)



= e

−2iπνa

f (ν), F



f (x) e

2πiν



f (ν −ν

These results should be compared to the formulas for similar operations on the

Fourier coefﬁcients of the periodic f unction; see Theorem 9.36 on page 264.

COROLLARY 10.2 3 In the following table, the properties of f indicated in the ﬁrst

column are transformed into the properties of

f in the second column, and conversely:

286 Fourier transform of functions

f (x)

f (ν)

even even

odd odd

real-valued hermitian

purely imaginary antihermitian

Here a f unction g is a hermitian func tion if g(−x) = g(x) and an antihermitian

function if g(−x) = −g(x).

10.2.b D ilation

If a dilation is performed (also called a change of scale), that is, if we put

g(x) = f (ax) ∀x ∈ R, for some a ∈ R

∗

the Fourier transform of g can be computed easily:

eg(ν) =

f ( ax) e

−2πiν x

dx =

f ( y) e

−2πiν y/a

|a|





which gives the following theorem.

THEOREM 10.24 Let a ∈ R

∗

and f ∈ L

. Then we have



f ( ax)



|a|





In other words, a dilat ion in the real world corresponds to a compression

in the Fourier world, and conversely.

10.2.c D erivation

There is a very important relation between the Fourier transform and the

operation of differentiation:

THEOREM 10.25 (Fourier transform of the derivative) Let f ∈ L

be a func-

tion which decays sufﬁciently fast at inﬁnity so that x 7→ x

f (x) belongs also to L

for k = 0, . . ., n. Then

f can be differentiated n times and w e have



(−2iπx)

f (x)



(k)

(ν) for k = 1, . . . , n.

Conversely, if f ∈ L

and if f is of C

class and, moreover, its successive

derivatives f

(k)

are integrable for k = 1, . . ., n, then we have



(m)

(x)



= (2iπν)

f (ν) for k = 1, . . ., n.