An introduction to

Fourier Analysis

University of Delhi

Professor Ajay Kumar

Department of Mathematics

Delhi-110007

There are several natural phenomena that are described by periodic functions. The position of a planet in its orbit around the sun is a periodic function of time; in Chemistry, the arrangement of molecules in crystals exhibits a periodic structure. The theory of Fourier series deals with periodic functions.

We begin with the concept of Fourier SeriesFourier Series Periodic function

A function f(t) is said to have a period T or to be periodic with

period T if for all t, f(t+T)=f(t), where T is a positive constant. The least value

of T>0 is called the principal period or the fundamental period or

simply the period of f(t).

The function has periods 2π, 4π, 6π,

all equal

.

Let

If f(x) has the period

then has the period T.

…… since

Let a functionf be declared on the interval [0,T). The periodic expansion

defined by the formula of f is

Periodic expansionPeriodic expansion

Theorem:

Let f be continuous on  

converges uniformly to f for all

.

Suppose that the series

 

Then

(2)

(1)

The numbers an and bn are called the Fourier coefficients of f. When an and

bn are given by (1)and(2) the trigonometric series is called the Fourier series of the

function f. Below is an example of an arbitrary function (the green function)

which we approximate with Fourier series of various lengths. As you can see, the

ability to mimic the behavior of the function increases with increasing series

length, and the nature of the fit is that the "spikier" elements are fit better by the

higher order functions.

In mathematics, the question of whether the Fourier series of a

function converges to the given function is researched by a field

known as classic harmonic analysis, a branch of pure mathematics. For most engineering uses of Fourier analysis,

convergence is generally simply assumed without justification. However, convergence is not

necessarily a given in the general case, and there are criteria which

need to be met in order for convergence to occur.

Find the Fourier series of the function

Hence

Example.

Find the Fourier series of the function

The Fourier series of f(x) is

Example.

Find the Fourier series of the function

Example.

would wonder how to define a similar notion for functions which are L-periodic. Assume that f(x) is defined and integrable on the interval [-L,L]. Set

Remark.

We defined the Fourier series for functions which are 2π -periodic, one

The function F(x) is defined and integrable on [-π , π ].Consider the Fourier

series of F(x)

Using the substitution,

we obtain the following definition:

Let f(x) be a function defined and integrable on [-L,L]. The Fourier series of f(x) is

.

Definition.

for

where

Find the Fourier series of

Example.

the Fourier series of f(x) can be written in complex form as

 

where

 

Using Euler's identities,

and

n =1,2,…

n =1,2,…

In what sense does the series on the right converge, and if it does converge, in what sense is it equal to f(x)? These questions depend on the nature of the function f(x). Considering functions f(x) defined on R which satisfy a reasonable condition like

| f(x) | dx < ∞

| f(x) |

2 dx< ∞

or

Since such functions cannot be periodic ,we cannot really hope to

expand such function in terms of sin kx and cos kx ,

k = 0,1, 2, --.As in the periodic case. Taking the clue

that the functions einx correspond to the homomorphism χn of S1 into S1 ,

we look for continuous homomorphism R into S1.It turns out that all such

continuous homomorphism are given by

φλ(x) = exp (iλx).

For an integrable function x(t) , define the Fourier transform by

every real number w.

The independent variable t represents time, the transform variable ω represents

angular frequency .Other notations for this same function are:    

 The function is complex-valued in general.  

and  

If is defined as above, and

is sufficiently smooth, then it can be

reconstructed by the inverse transform:

for every real number t

.

For a scalar random variable X the characteristic function is defined as the expected value of eitX, and t ∈ R is the argument of the characteristic function:

Here FX is the cumulative distribution function of X, If

random variable X has a probability density function ƒX,

then the characteristic function is its Fourier transform.

In this section, all the results are derived for the following definition (normalization)

of the Fourier transform:

Let's see how we compute a Fourier Transform: consider a particular function

f(x) defined as

Its Fourier transform is:

In this case F(u) is purely real, which is a

consequence of the original data being

symmetric in x and -x. A graph of F(u) is

shown in Fig.

Fourier Transform--Gaussian

The Fourier transform of a Gaussian

function is given by

The second integrand is odd, so integration over a symmetrical range gives 0.The value of the first integral is given by Abramowitz and Stegun ,so

a Gaussian transforms to another Gaussian

The Fourier transform of the Gaussian function is another Gaussian:

Note that the width sigma is oppositely positioned in the arguments of the exponentials. This means the narrower a Gaussian is in one domain, the broader it is in the other domain. The Fourier transform can also be extended to the space integrable functions defined on

where,

and is the space of continuous functions on

.

In this case the definition usually appears as

and

is the inner product of the two vectors ω and x.One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in

where

The Plancherel theorem and Parseval's theorem

It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.If f(t) and g(t) are square-integrable and F(ω) and G(ω) are their Fourier transforms, then we have

Parseval's theorem:

.

where the bar denotes complex

conjugation. Therefore, the Fourier

transformation yields an isometric

automorphism of the Hilbert space

The Plancherel theorem, a special case of Parseval's theorem, states that

This theorem is usually interpreted as asserting the unitary property of the Fourier transform.

For reasonable functions f and h , we define the convolution f * h by

The convolution is explained by the following graphs

Convolution of Functions

then

1. ( a f + bg )^ = a f^ + b g^ where a,b ε C

2. If g (x) = f (x+ u), then g^(y) = exp( 2πi yu) f^(y)

3.If h(x) = exp(2πi ux), then h^(y) = f^(y-u)

Basic Facts about the Fourier transform

For reasonable functions

4. (f ΄)^(y) = 2πi y f^ (y) ,

where f ΄ is the derivative of f

The Fourier transform has become a powerful analytical

tool in diverse fields of science. In some cases, the Fourier

transform can provide a means of solving unwieldy equations

that describe dynamic responses to electricity, heat or

light.

In other cases, it can identify the regular contributions to a

fluctuating signal, thereby helping to make sense of

observations in astronomy, medicine and chemistry. Perhaps because of its usefulness, the Fourier

transform has been adapted for use on the personal computer.

Consider the heat flow in an infinite rod where the initial temperature is given . In other words, we look for the solution to the initial –value problem, sometimes called a Cauchy problem PDE 2 , 0t xxu u x t

IC ( ,0) ( )u x x x

Application of Fourier transform: Solution of an Initial -Value

Problem

There are three basic steps in solving this problem.

STEP 1(Transforming the problem)

Since the space variable x ranges from

transform of the PDE and IC with respect to this variable

x . Doing this, we get

, we take the Fourierto

2[ ] [ ]

[ ( ,0)] [ ( )]t xxF u F u

F u x F x

and using the properties of the Fourier transform, we have

2 2( )( )

dU tU t

dt

(0) ( )U is the Fourier transform of

( ) [ ( , )]U t F u x t

where

is nothing more than a constant in this differential equation, so the solution to

this problem is

2 2

( ) ( ) tU t e

Step 2 (Solving the transformed problem)

Remember the new variable

Step 3 (Finding the inverse

transform)

To find the solution u(x,t) , we merely compute

2 21 1( , ) [ ( , )] [ ( ) ]tu x t F U t F e

Now using one of the properties of convolution, we get

2 21 1( , ) [ ( )] [ ]tu x t F F e 2 2( / 4 )1

( )2

x tx et

2 2( ) / 41( )

2x te d

t

(using tables)

This is the solution to our problem.

2

21

1

6n n

( )f x x (0,2 )

2as

Let in

-periodic.

22

2( )f n f

2 212

2 21 1

1 1 1(0) 2

n n

fin n n

2 22 2

20

1 4

2 3f x dx

2 22

21

1 42

3 3n n

2

21

1

6n n

So,

Consider nth order linear nonhomogeneous ordinary differential equations with constant coefficients

11 1 0

( ) ( )n n

n n

Ly x f x

L a D a D a D a

dD

dx are constants,

, 0,1, ,ia i n

Taking Fourier transform

11 1 0

ˆˆ[ ( ) ( ) ( ) ] ( ) ( )n nn na ik a ik a ik a y k f k

where

0

( )n

rr

r

P z a z

ˆ ( ) ˆ ˆˆ ˆ( ) ( ) ( ) ( )( )( )

f ky k f k q k f g k

P ik

where

1ˆ( )

( )q k

P ik

( ) ( ) ( )y x f q x d

ˆˆ( ) ( ) ( )P ik y k f k

Find the solution of the ordinary differential equation

22

2( ),

d ua u f x x

dx

Applying Fourier transform

2 2

ˆ ( )ˆ( )

( ) ( ) ( )

f ku k

k a

u x f g x d

where

12 2

1 1( ) ( ) exp( )

2g x f a x

k a a

so ( ) ( ) a xu x f e d

Solve the following ordinary differential equation

'' '2 ( ) ( ) ( ) 0u t tu t u t

Applying Fourier transform

2 'ˆ ˆ2 ( ( )) 0d

w u i f u t udw

Or

2 ˆ ˆ ˆ2 (( ) ( )) 0d

w u i i wu w udw

2

ˆˆ2

ˆ( ) w

duwu

dw

u w ce

2

ˆˆ2

ˆ( ) w

duwu

dw

u w ce

Inverse Fourier transformation gives

2( )4( )t

u t De

The method of Fourier transform can be used to solve integral

equations

( ) ( ) ( ) ( )f t g x t dt f x u x

Where g(x)and u(x) are given functions and is a known parameter.Applying Fourier transform

ˆ ˆˆ ˆ( ) ( ) ( ) ( )f k g k f k u k

ˆ( )ˆ ( )ˆ ( )

ˆ( )( )

ˆ ( )ikx

u kf k

g k

u kf x e dk

g k

In particular, if so that then the solution

becomes

1( )g x

x

ˆ ( ) sgng k i k

ˆ1 ( )( )

2 ( sgn )

ikxu k ef x dk

i k

If , and so that , 1( )

2

xg x

x

1ˆ( ) ,g k

ik

the solution reduces to

'ˆ1 ( ) 1( ) ( ( )) ( )

2 (1 ) 2

ikxx ikxu k e

f x dk f u x f e e dkik

' '( ) ( ) exp( )xu x e u x

Consider the solution of the Laplace equation in the half plane

0, y 0xx yyu u x

with the boundary conditions u(x,0)=f(x), x

( , ) 0u x y ,x y as

Apply Fourier transform with respect to x, to obtain 2

2ˆˆ 0

d uku

dy

ˆˆ ˆ( ,0) ( ), ( , ) 0u k f k u k y as y

ˆˆ( , ) ( ) exp( )u k y f k k y

( , ) ( ) ( )u x y f k g x d

where1

2 2

1( ) ( )k y yg x f e

x y

( )

( ) sin

xf x e

f x x

1[ ] ( )

2i xF f f x e dx

x

The major drawback of the Fourier transform is that all functions can not be transformed; for example, even simple functions like

cannot be transformed, since the integral

does not exist. Only functions that damp to zero sufficiently fast ashave transforms.

As a rule of thumb;the more

concentrated f(t) is, the more spread

out is F(ω). In particular, if we

"squeeze" a function in t, it spreads out

in ω and vice-versa; and we cannot

arbitrarily concentrate both the function

and its Fourier transform.

An atom, and its Fourier Transform:

Note the both functions have circular

symmetry. The atom is a sharp feature, whereas its transform is a

broad smooth function.This illustrates

the reciprocal relationship between a

function and its Fourier transform.

A molecule, and its Fourier Transform:

The molecule consists of seven atoms. Its transform shows some

detail, but the overall shape is still that of the atomic transform. We can consider the molecule as the

convolution of the point atom structure and the atomic shape. Thus its transform is the product of the point atom transform and

the atomic transform

If we think of concentration in terms of f living entirely on a set of finite measure,then we have the following beautiful result of Benedicks: Let f be a nonzero square

integrable function on R. Then the Lebesgue measures of the sets { x, f(x) ≠ 0 } and

{y, f^(y) ≠ 0 } cannot both be finite.

Benedicks’s theorem

( For those who are not familiar with the jargon of measure theory, a (measurable) subset A of R is of finite measure, if it can be covered by a countable union of intervals Ik such that ( length of Ik) < ∞ .). This result is a significant generalization of the fact , well known to engineers, that a nonzero signal cannot be both time limited and band limited.

k

The rate at which a function decay at infinity can also be considered a measure of concentration. The

following elegant result of Hardy’s states that both f and f^ cannot be

very rapidly decreasing: Suppose f is a measurable function on R such that

│f(x) │≤ A exp(-απx2 ) and

│f^(y) │≤ B exp(-βπy2 ) for some positive constants A,B, α,β then, if

Hardy’s theorem:

i αβ > 1,then f must necessarily be a zero function a.e.

ii αβ < 1, then there are infinitely many linearly independent functions

iii αβ = 1, then f(x) = c exp(-απx2) for some constant c.

For f ε L1( R ),

R R

implies f = 0 a.e.

Beurling’ Theorem

│f(x) ││f^(y) │exp( 2π │xy│) dx dy < ∞

Generalizations of these results have been obtained for a variety of locally compact groups like Heisenberg groups, Motion groups, non-compact connected semi-simple Lie groups and connected nilpotent Lie groups

Jean Baptiste Joseph Fourier(1768-1830)

Administrator, Egyptologist, engineer, mathematician, physicist,

revolutionary, soldier, and teacher; incredible as it may sound, Fourier

was all these! Born on March 21, 1768 in Auxerre, France, orphaned at the

age of ten, Fourier had a brilliant school career in the local Benedectine

school in his home town.

His life was rich and varied: member and president of the local revolutionary

committee during his youth, student at the Ecole Normale, teacher at the Ecole Polytechnique (where he succeeded the great Lagrange in the chair of Analysis

and Mechanics), imprisoned several times thanks to rapidly shifting political

ideologies in Paris, and finally as recognition of his distinguished service to science, he was elected permanent mathematical secretary of the French

Academy of Sciences in 1822

His mathematical and science achievements are, of course, legion. He derived the partial differential equation that governs heat conduction, and, to study the problem of heat conduction, systematically developed the subject

which later came to be known as Fourier analysis- a subject which was in its

infancy during his time. Far transcending the particular subject of heat conduction,

his work stimulated research in mathematical physics, which has since

been often identified with the solution of boundary-value problems,

encompassing many natural occurrences such as sunspots, tides, and the

weather. His work also had a great influence on the theory of functions of a real variable, one of the main branches of modern mathematics. Some of his great discoveries are contained in his

celebrated treatise ‘Théorie analytique de la chaleur’ published in 1822. Fourier

also made important contributions to probability theory, statistics, mechanics,

optimization and linear programming.

G.B.Folland and A.Sitaram ( J.Fourier Anal.Appl.(1997)

G.F.Price and A.Sitaram ( J.Functional Analysis (1988))

E.Kaniuth and A.Kumar ( Forum Mathematicum (1991))

E.Kaniuth and A.Kumar( Math. Proc.Camb. Phil.Soc.(2001)

S.Thangavelu (Colloquium Mathematicum (2001) and Math.Zeit.(2001)

H.Dym and H.P.McKean, Fourier Series and Integrals,Academic Press,1972.

R.Bhatia, Fourier Series, TRIM-2.Hindustan Book Agency, 1993.

T.W. Korner,Fourier Analysis, Cambridge University Press, U.K., 1988.

G.B.Folland, Fourier Analysis and its Applications, Wadsworth and Brooks/ Cole, U.S.A., 1992

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