Cal3-Dinh_Hai (09-10) FourierSeries SLIDES

8/10/2019 Cal3-Dinh_Hai (09-10) FourierSeries SLIDES

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Chapter 4 FOURIER SERIES

Contents

4.1 Introduction

4.2 Fourier Series Expansion

4.3 Functions Defined over a Finite Interval

4.4 Differentiation and Integration of Fourier Series

4.5 Engineering Applications

4.6 Complex Fourier Series4.7 Orthogonal Functions

Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
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Chapter 4 FOURIER SERIES4.1 INTRODUCTION

The representation of a function in the form of a series is fairlycommon practice in mathematics. Probably the most familiarexpansions are power series of the form

f(x) =a0+a1x+ +anxn + =n=0

anxn.

There are frequently advantages in expanding a function in such aseries, since the first few terms of a good approximation are easily

to deal with.The functions represented by power series play important role inanalysis. But the class of these functions is too restricted in manyinstances.

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4.1 INTRODUCTION

It was therefore an even of major importance when Fourier

illustrated by many examples the fact that convergenttrigonometric seriesof the form

f(x) =a0

2 +

n=1(ancos nx+bnsin nx)

with constant coefficients an, bn are capable of representing a wideclass of functions f(x) which includes essentially every function ofspecific interest.

Soon after Fouriers dramatic discovery the Fourier series wererecognized not only as a most powerful tool for physics andmathematics, but just as much as a fruitful source of manybeautiful purely mathematical results.

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4.1 INTRODUCTION

There are also many practical reasons for expanding a function asa trigonometric sum. Iff(t) is a signal, then a decomposition offinto a trigonometric sum gives a description of its componentfrequencies.

A common task in signal analysis is the elimination of highfrequency noise. One approach is to express f as a trigonometricsum

f(t) =a0

2 +

n=1(ancos nt+bnsin nt)

and then set the high-frequency coefficients (the an and bn forlarge n) equal to zero.

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4.1 INTRODUCTION

Another common task in signal analysis is data compression.

The goal here is to send a signal in a way that requires minimadata transmission. One approach is to express the signal f in terms

of a trigonometric expansion and then send only those coefficients,an and bn, that are larger (in absolute value) than some specifictolerance. The coefficients that are small and do not contributesubstantially to f can be thrown away.

There is no danger that an infinite number of coefficients staylarge, because we will show that an 0 and bn 0 as n .

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4.1 INTRODUCTION

In general, as mentioned in previous chapter, periodic functionsfrequently occur as input signals in engineering applications.

Fourier series provide the ideal framework for analysing the

steady-state response to such periodic input signals, since theyenable us to represent the signals as infinite sums of sinusoids.Because of the linear character of the system, the desiredsteady-state response can be determined as the sum of the

individual responses.

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4.1 INTRODUCTION

Functions that are expressible by trigonometric series

f(t) = a0

2 +

n=1

(ancos nt+bnsin nt)

are periodic with the period 2/. But, as we shall see, thisrestriction is inessential as soon as we consider a function merely ina finite interval from which we can easily extend it as a periodicfunction.

The theory of Fourier series is complicated, but we shall see thatthe application of these series is rather simple. The fundamentalquestion in this chapter is devoted to concern the problem ofrepresenting an arbitrary periodic function in the form of a sum ofsine and cosine functions.

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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS

Recall that a function f(x) is called a periodic function if there issome positive number T, called a period off(x), such thatf(x+T) =f(x) for all x in the domain off.

If f is a continuous periodic function which is not identically equalto a constant, it possesses the least positive period called itsfundamental period.

When speaking about the period of a function we usually mean itsfundamental period.

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To provide a measure of the number of repetitions per unit ofx,

we define the frequency of a periodic function to be the reciprocalof its period, so that

frequency = 1

period=

1

T

The term circular frequency is also used in engineering, and isdefined by

circular frequency = 2

frequency =2

T

and is measured in radians per second.

It is common to drop the term circular and refer to this simplyas the frequency when the context is clear.


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Periodic Extension of a Function

Let f(x) be any function defined on [a, b]. Set T =b a. Thefunction defined by

(x) =

f(x) if a


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Integral of a Periodic Function over a Period

Iff(x) is a periodic integrable function with periodic Twe have

+T

f(x)dx=

T0

f(x)dx for any < <

Thus, the integral of a periodic function with period T taken overan arbitrary interval of length T has one and the same value.


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In physics the simplest periodic processes are described by means

of the function

(t) =A sin(t+ ),


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The fundamental question in this section is devoted to concern theproblem of representing an arbitrary periodic function in the formof a sum of harmonics:

f(x) =A0+A1sin(x+1)+A2sin(2x+2)+ +Ansin(nx+n) .(1)

The term A1sin(x+ 1) is called the first harmonic or thefundamental mode, and it has the same frequency as thefunction f(x).

The term Ansin(nx+ n) is called the nth harmonic, and it hasfrequency n, which is n times that of the fundamental.


4 2 FOURIER SERIES EXPANSION
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The expression (1) may be written as

f(x) =a0

2 +

n=1

ancos nx+bnsin nx

. (2)

The functional series on the right hand side of (2) is called atrigonometric series.


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Statement of the Key Problem The main aim of the present

chapter is to elucidate the following questions:

(1) What are the periodic functions with period 2lwhich can beexpressed in trigonometric series

f(x) =a0

2 +

n=1

ancos

nx

l +bnsin

nx

l

, (3)

i.e., can be represented as a sum of this kind?

(2) How can we determine the coefficients a0, an and bn,n= 1, 2,... of expression (3) if it is valid?


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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES

Recall that

A functionf(x) is said to have a jump discontinuity atx0 (a, b) iff(x) is discontinuous at x0 and the one-sided limits

limxx0

f(x) and limxx+0

f(x)

exist as finite numbers. If the discontinuity occurs at an endpoint,x0 =a (orb), a jump discontinuity occurs if the one-sided limit off(x) as x a+ (x b) exists as a finite number.

A functionf is said to be piecewise continuous on [a, b] if it

is continuous at every point in [a, b] except possibly for a finitenumber of points at which fhas a jump discontinuity.

A functionf is said to be odd iff(x) =f(x) for all x in thedomain off. f is even iff(x) =f(x) for all x in the domain off.


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Knowing that a function is even or odd can be useful in evaluating

definite integrals.

Theorem

If f is an even piecewise continuous function on [

a, a], then a

a

f(x)dx= 2

a0

f(x)dx.

If f is an odd piecewise continuous function on [a, a], then aa

f(x)dx= 0.


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Example 2.1 Show that

(a)

l

lsin mx

l cos nx

l dx= 0.

(b)ll

sin mxl

sin nxl

dx=

0, m =n,l, m=n.

(c)ll

cos mxl

cos nxl

dx=

0, m =n,l, m=n.


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It is easy to verify that if each of the functions f1, f2,..., fn isperiodic with period T, then so is any linear combinationc1f1(x) +c2f2(x) + +cnfn(x).

For example, the sum 5 + 3 sin x 8sin3x+ 7 cos 3x hasperiod 2 since each term has period 2.

Furthermore, if the infinite series

a0

2 +

n=1

ancosnxl

+bnsinnx

l

converges to f(x) for all x, then f(x) will be periodic with period2l.


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Let f(x) be an arbitrary integrable function defined on [l, l].

DefinitionThe numbers a0, an and bn, n= 1, 2, ..., determined by

an =1

l l

l

f(x)cosnx

l dx, n= 0, 1, 2, ..., (4)

bn =1

l

ll

f(x)sinnx

l dx, n= 1, 2, ..., (5)

are called the Fourier coefficients of the function f(x). The series

a02

+n=1

ancos

nx

l +bnsin

nx

l

is called the Fourier series of the function f(x).


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Formulae (4)(5) are called the Euler formulae for the Fourier

coefficients an and bn.

Note Iff(x) is a periodic function of period T = 2/ then itsFourier coefficients can be given by

an = 2

T

+T

f(x)cos nxdx (n= 0, 1, 2, . . . )

bn = 2

T

+T

f(x)sin nxdx (n= 1, 2, . . . )

The limits of integration in these formulas may be specified overany period. In practice, it is common to specify f(x) over eitherthe period

12 T

x

12 Tor the period 0

x

T.


4.2 FOURIER SERIES EXPANSION


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To every function f(x) integrable on the interval [l, l] therecorresponds its Fourier series. This correspondence can be writtenin the form

f(x) a02

+

n=1

ancosnx

l +bnsin

nx

l

where the series on the right-hand side is the trigonometric serieswhose coefficients are determined by Euler formulas.

Example 2.2 Obtain the Fourier series of the periodic functionf(t) of period 2 defined by

f(t) =t (0< t


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4.2 FOURIER SERIES EXPANSION4.2.3 FOURIER SERIES OF EVEN AND ODD FUNCTIONS

Theorem

(a) The Fourier series for an even function is a pure cosine series;that is, it contains no sine terms.

(b)The Fourier series for an odd function is a pure sine series; thatis, it contains no cosine terms.

Example 2.3 Letf(x) =x,l


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S S S4.2.4 CONVERGENCE OF FOURIER SERIES

Let

f(x+ 0) = limh0+ f(x+h) and f(x 0) = limh0f(x h).The following theorem covers most of the situations that arise inapplications.

Theorem(Pointwise Convergence of Fourier Series)If f and f are piecewise continuous on [l, l], then on (l, l), theFourier series of f

a0

2 +

n=1

ancosnxl

+bnsinnx

l

converges to f(x) if f is continuous at x, and to12 [f(x+ 0) +f(x 0)] if f is discontinuous at x . For x= l , theseries converges to 12 [f(

l+ 0) +f(l

0)].


4.2 FOURIER SERIES EXPANSION
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4.2.4 CONVERGENCE OF FOURIER SERIES

In other words, when f and f are piecewise continuous on [

l, l],

the Fourier series converges to f(x) whenever f is continuous at xand converges to the average of the left- and right-hand limits atpoints where f is discontinuous.

Example 2.5 Compute the Fourier series for

f(x) =

0,


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4.2.4 CONVERGENCE OF FOURIER SERIES

Example 2.7 Find the expansion of the function

f(x) = |x|, x into Fouriers series

Remark: It can be proved that iff is piecewise continuous and

f(x) =c0

2 +

n=1

cncos

nx

l +dnsin

nx

l

on [

l, l], then this series must be the Fourier series off.

Therefore, a function fcan be expanded in one, and only one,Fourier series on the interval [l, l].

Example 2.8 Find the Fourier series of the functionf(x) = cos2 xon the interval [

, ].


4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL
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4.3.1 FULL-RANGE SERIES

Consider a function fdefined in an interval [0, l] and suppose thatit is required to expand f into a trigonometric series.


To obtain a full-range Fourier series representation off(x) (that is,a series consisting of both cosine and sine terms) we define theperiodic extension (x) off(x) by

(x) =

f(x), if 0


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The Fourier coefficients are

an =2l

l

0f(x)cos2n

l xdx (n= 0, 1, 2, . . . )

bn =2

l

l0

f(x)sin2n

l xdx (n= 1, 2, . . . )

Example 2.9 Find a full-range Fourier series expansion off(x) =xvalid in the finite interval 0< x< . Draw graphs ofboth f(x) and the periodic function represented by the Fourier

series obtained.

In applications the most important role is played by the expansionsinto Fouriers sine and cosine series.


4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL
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A functionf(x) can also be extended to the interval [

l, 0) in such

a way that the extended function F(x) is even. This isaccomplished by defining

F(x) = f(x), 0 x l,f(

x),

l

x

0.

Since F is even, it is called the even extension off. Thus, theeven periodic extension off(t) is defined by

F(x) = f(x), 0 x l,f(x), l x 0

F(x+ 2l) =F(x).


4.3 FUNCTIONS DEFINED OVER A FINITE INTERVALG
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4.3.2 HALF-RANGE COSINE AND SINE SERIES

Similarly, the function f(x) can be extended to the interval [l, 0)as an odd function defined in the interval [

l, l]. The function

G(x) =

f(x), 0


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Definition

Let fbe piecewise continuous on the interval [0, l]. The Fouriercosine series off on [0, l] is

a02 +

n=1

ancosnx

l , (6)

where

an =2

l l

0f(x)cos

nx

l dx, n= 0, 1, 2,....


4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL4 3 2 HALF RANGE COSINE AND SINE SERIES
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Definition

The Fourier sine series off on [0, l] is

n=1bnsin

nx

l , (7)

where

bn =2

l

l0

f(x)sinnx

l dx, n= 1, 2,....

(6) and (7) are called half-range Fourier series expansions forf(x). (6) is half-range cosine series expansion and (7)half-range sine series expansion off(x).


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TheoremIf f and f are piecewise continuous on [0, l], then on this interval,f(x) can be expanded in the Fourier cosine series

a02 +

n=1

ancos

nx

l

or in the Fourier sine series

n=1

bnsin

nx

l .


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To illustrate the various extensions, lets consider the functionf(x) =x, 0

x

.

The period extension (x) is given by

(x) =f(x) =x, 0< x<

(x+ ) = (x).

Then the full-range Fourier series representation off(x) is

2

n=1

1

nsin 2nx,

which consists of both odd functions (the sine terms) and evenfunction (the constant term) since the extension (x) is neither aneven nor an odd function.


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The even extension off is the functionF(x) =

|x|,

x

.Thus, the half-range cosine series expansion for f(x) is

2 4

n=11

(2n 1)2 cos(2n 1)x.

The odd extension off is G(x) =x,


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The preceding three extensions are natural extensions off. Thereare many other ways of extending f(x). For example, the function

H(x) =

0, x


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4.4.1 DIFFERENTIATION OF A FOURIER SERIES

The term-by-term differentiation of a Fourier series is not alwayspermissible. The following theorem gives sufficient conditions for

using termwise differentiation.

Theorem(Differentiation of Fourier Series)

Let f be a continuous function on(,) and periodic of period2l. Let f(x) and f(x) be piecewise continuous on [

l, l]. Then,

the Fourier series of f can be obtained from the Fourier series forf by termwise differentiation. In particular, if

f(x) =a0

2 +

n=1

ancosnx

l +bnsin

nx

l ,then

f(x) n=1

n

l

ansinnx

l +bncos

nx

l

.


4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4 4 1 DIFFERENTIATION OF A FOURIER SERIES
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4.4.1 DIFFERENTIATION OF A FOURIER SERIES

Example 4.1 Consider the process of differentiating term byterm the Fourier series expansion of the function

f(t) =t2, t f(t+ 2) =f(t).


4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4.4.2 INTEGRATION OF FOURIER SERIES
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4.4.2 INTEGRATION OF FOURIER SERIES

Theorem(Integration of Fourier Series)

Let f be piecewise continuous on [l, l] with Fourier series

f(x)

a0

2

+

n=1

ancosnxl

+bnsinnx

l,

then, for any x in [l, l], we have

x

l

f(t)dt= x

l

a0

2dt+

n=1

x

lancosnt

l +bnsin

nt

ldt.


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Example 4.2 The Fourier series expansion of the function

f(t) =t2

( t ), f(t+ 2) =f(t)is

t2 =2

3 + 4

n=1(1)n cos nt

n2 , t .

Integrating this result between the limits and x gives x

t2dt=

x

2

3dt+ 4

n=1

x

(1)n cos ntn2

dt

that is,

1

3x3 =

2

3x+ 4

n=1

(1)n sin nxn3

, x .




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Rearranging, we have

x3 2x= 12n=1

(1)n sin nxn3

, x .

and the right-hand side may be taken to be the Fourier seriesexpansion of the function

g(x) =x3 2x, x

g(x+ 2) =g(x).


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Example 4.3 Use the Fourier series

x2 =2

3 + 4

n=1(1)n

n2 cos nx, x

to deduce the value of the series

k=1(1)k+1

(2k 1)3 .


4.5 ENGINEERING APPLICATIONS
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We know that forced oscillations of a body of mass m on a springof modulus kare governed by the differential equation

my

+cy

+ky=r(t) (8)

where y(t) is the displacement from rest, cthe damping constant,kthe spring constant (spring modulus), and r(t) the external forcedepending on time t.

In (8), letm= 1 (gm), c= 0.05 (gm/sec), and k= 25 (gm/sec2),so that (8) becomes

y + 0.05y + 25y=r(t)

where r(t) is measured in gmcm/sec2. Let

r(t) =

t+ 2 , if


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Consider a Fourier series

f(x) = a02 +

n=1

ancos nx+bnsin nx

.

Let us replace cos nxand sin nxby their expressions

cos nx= e

jnx

+e

jnx

2 , sin nx= e

jnx

ejnx

2j .

Then

f(x) =

a0

2 +

n=1

an

ejnx +ejnx

2 +bnejnx

ejnx

2j

= a0

2 +

1

2

n=1

ejnx(an jbn) +ejnx(an+jbn)

. (9)


4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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Denote cn = (an jbn)/2 and use Eulers formulas for an and bnto obtain:

cn = an jbn

2 = 1

2

f(x)cos nxdxj 1

2

f(x)sin nxdx

= 1

2

f(x)

cos nxjsin nx

dx.

Thus,cn =

1

2

f(x)ejnxdx.

Let

cn = cn =an+jbn

2=

1

2

f(x)

cos nx+jsin nx

dx

= 1

2

f(x)ejnxdx.


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We now write down series (9) in the form

f(x) =c0+n=1

cnejnx +

n=1

cnejnx

or, briefly,

f(x) =

n=

cnejnx where cn =

1

2

f(x)ejnxdx (10)

(10) is the so-called complex form of the Fourier series, morebriefly, the complex Fourier series, off(x).

Thecn are called the complex coefficients off(x).


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The sum

cnejnx +cne

jnx =cnejnx +cnejnx = 2Recnejnx

=Re

(an jbn)(cos nx+jsin nx)

=ancos nx+bnsin nx.

is a real function representing the nth harmonic. It can be

rewritten as

ancos nx+bnsin nx=Ancos(nxn)where An =

a2n+b

2n = 2|cn| is the amplitude; since tan n = bnan ,

the initial phase is expressed as n =Argcn =

Arg cn.

The sequence of complex numbers cn is called the spectralsequenceof the function f(x); the real sequence 2|cn| is termedthe amplitude spectrumoff(x) and the sequence n = Arg cnthe phase spectrum.


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In the general case we deal with a function f(x) defined in aninterval [

l, l] for which the Fourier trigonometric expansion is

constructed. In this case Fourier series is written in complex formas

f(x) =

n=

cnejnx where cn =

1

2l l

l

f(x)ejnxdx and n =n

l

Iff(x) is a periodic function of period T its complex form of theFourier series expansion is

f(x) =

n=

cnejnx where cn =

1

T

+T

f(x)ejnxdx and n =2n

T


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Example 6.1 Find the Fourier series of the functionf(x) =ex

on the interval [, ].


4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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Theorem

Let f and g be piecewise continuous functions on[l, l]. If an, bnare the Fourier series coefficients of f(x) andn, n those of g(x),

then

1

2l

ll

f(x)g(x)dx=1

40a0+

1

2

n=1

(nan+ nbn)


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In terms of complex coefficients the multiplication theorem can bestated as follows.

If f and g are piecewise continuous functions on[l, l], then

12l

l

l

f(x)g(x)dx=

n=

cndn

where the cn and dn are the coefficients in the complex

Fourier series expansion of f and g respectively.


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If we put g(x) =f(x) in the above result, the following importanttheorem immediately follows.

Theorem(Parsevals theorem)

If f is piecewise continuous on[l, l] and has Fourier coefficientsan, bn, then

1

2l

ll

[f(x)]2dx=1

4a20+

1

2

n=1

(a2n+b2n) (11)

If cn are the coefficients in the complex Fourier series expansion off then

1

2l

ll

[f(x)]2dx=

n=

|cn|2


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Parseval equality (11) implies that the series

a202

+n=1

a2n+b

2n

is convergent and, consequently, we have

limn

an =1

l limn

ll

f(x)cosnx

l dx= 0

and

limn

bn =1l

limn

l

l

f(x)sinnxl

dx= 0.


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Example 6.2 Given the Fourier series

t2 =2

3 + 4

n=1

(1)nn2

cos nt, t

deduce the value ofn=1

1

n4.


4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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Definition

Two real functions f(x) and g(x) that are piecewise continuous in

the interval [a, b] are said to be orthogonal in this interval if

ba

f(x)g(x)dx= 0.

A set of functions

1(x), 2(x), . . . , n(x), . . .

each of which is piecewise continuous on [a, b], is said to be an

orthogonal set on this interval if ba

m(x)n(x)dx=

0 if m =n>0 if m=n.


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Definition

An orthogonal set{

n(x)}

is said to be orthonormal if

ba

n(x)

2dx= 1 for all n= 1, 2, . . .

Example 7.1 (a) The set

1

2, cos

x

l , sin

x

l , cos

2x

l , sin

2x

l , . . . , cos

nx

l , sin

nx

l , . . .

is orthogonal on [l, l]. The set12l

, 1

lcos

x

l ,

1l

sinx

l ,

1l

cos2x

l ,

1l

sin2x

l , . . . ,

1l

cosnx

l ,

forms an orthonormal set on the same interval.Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES

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(b) The system ofLegendre polynomials

P0(x) = 1, Pn(x) = 1

2nn!

dn

dxn

(x2 1)n, n= 1, 2, . . .is orthogonal on [1, 1], while the set

12

P0(x),

2n+ 1

2 Pn(x), n= 1, 2, . . .

is an orthonormal set on [1, 1].


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If a function (x) is piecewise continuous on [a, b], thenonnegative number

=

b

a

2(x)dx

1/2

is called the normof(x) on [a, b].

For example, on [l, l],

1

2

=

l

2,cosnx

l

= l, cosnxl

= l, n= 1, 2, . . .The norms of Legendres polynomials on the interval [

1, 1] are

Pn(x) = 1

1P2n(x)dx

1/2=

2

2n+ 1, n= 0, 1, 2, . . . .

We note that any orthogonal set{n(x)} can be converted intoan orthonormal set by diving each member n(x)ofitsnorm.


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Examples of other sets of orthogonal functions that are widelyused in practice are Bessel functions, Hermite polynomials,Laguerre polynomials, Jacobi polynomials, Chebyshev polynomialsand Walsh functions.

Over recent years wavelets are another set of orthogonalfunctions that have been widely used, particularly in applicationssuch as processing and data compression.


4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
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Definition

Iff(x) is a piecewise continuous function on the interval [a, b] and{n(x)} is an orthogonal set on this interval then the numbers

cn = 1

n2 ba

f(x)n(x)dx, n= 1, 2, . . .

are called the generalized Fourier coefficients of the functionf(x) with respect to the basic set{n(x)}. The series

n=1 cnn(x)

is called the generalized Fourier series off(x) with respect tothe basic set{n(x)}.



L f ( ) b i i i [ b] A f i f h f
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Let f(x) be piecewise continuous on [a, b]. A function of the form

F(x) =N

k=1

nk(x),

where 1, 2,...,n are constants, can be considered anapproximation of the given function f(x). F(x) is called a

polynomial of order Nwith respect to orthogonal system{n(x)}.

Question: Is the partial sum

SN(x) =N

n=1

cnk(x)

of the generalized Fourier series the bestapproximationoff?.Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES


W d fi th E b t th t l l f
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We define the mean square error ENbetween the actual value off(x) and the approximation F(x) as

EN= 1b a

b

a

f(x) F(x)2dx.

Clearly, EN 0.

Theorem(Minimum Square Error)The square error of F(x) =

Nn=1 nn(x) (with fixed N) relative

to f on[a, b] is minimum if and only if the coefficients of F are thegeneralized Fourier coefficients of f . This minimum value is givenby

EN=

ba

f2dxN

n=1

c2nn2 (12)

where cn are the generalized Fourier coefficients of f .


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The polynomialSN(x) =

Nn=1

cnk(x)

whose coefficients cn are the generalized Fourier coefficients off

with respect to the given orthogonal system{n(x)} is called theFourier polynomial of the function fwith respect to theorthogonal system{n(x)}.

Thus, the mean square error of polynomial F(x) relative to f isminimum if and only if F is the Fourier polynomial of f .


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Since EN 0 and (12) holds for every N, we obtain Besselsinequality

n=1

c2nn2 ba

f2(x)dx (13)



O i h i i i i h h E 0
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One question that arises in practice is whether or not EN 0 asN

. If this were the case then

n=1

c2nn2 = ba

f2(x)dx (14)

which is the generalized form of Paservals equality, and the set{n(x)} is said to be complete.

In the case of the trigonometric system equality (14) turns into

a202

+n=1

a2n+b

2n

=

1

l

ll

f2(x)dx

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Documents

Cal3-Dinh_Hai (09-10) FourierSeries SLIDES