Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations

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FOURIER SERIES

arbitrary linear combination of the orthonormal functions φ

(χ) represented by

Ajv(x)

= ^2α

(χ)

n=l

the mean-square error is

E=\\f-

Λ||

= f [/(*) -

(x)}

(2.36)

Note: If a set of coefficients {7

} minimizes the mean-square error E, the same set

minimizes (see Exercise 2.6)

||/(x)

AJV(X)||

= Jj* \f{x) -

AN(X)}

The integrand on the right-hand side of Equation (2.36) must be expanded before

the integration is carried out. To that end

[/(*) -

(x)}

= (f(x)f - 2f(x)A

(x) + (A

(x)f (2.37)

and the last term on the right-hand side of Equation (2.37) may, in turn, be expanded

(ΛΛΓ(Χ))

= A

(x)A

(x)

I ]Γ)

™Φπι(χ) 1

I Σ

ηΦη(χ)

\ra=l

) \n=l /

= I 5Z

τηΦπι(χ) 1 (αιφι(χ) + «202^)

l· αΝφΝ(χ))

\m=l

</>

(x) αι0ι(χ) + J^ (x) α

{χ) + ··· +

m=:l

/ \ra=l /

\ I

(a;)

(χ)

(/)

(x)

(χ)

Vm=l

n=l

\ra=l

/ N \

Σ I Σ

πιΦπι(χ)θίηΦη(χ)

n=l

\m=l /

/ N \

5Z ( Σ

™

η0™(ζ)<Μ#) I

(2.38)

BEST APPROXIMATION 43

Substituting the result in Equation (2.38) into Equation (2.37) gives

N / N \

[f(x)-A

(x)}

= (¡(χ))

-2/(χ)Α

(χ) + ^2ΐγ^α

πι

(χ)φ

(χ)\

n=l \m=l /

= (ί(χ))

-2/(χ)\Υ^α

(χή +

N / N

Σ ( Σ

πιΟίηΦπι(χ)Φη(χ)

n=l \ra=l >

(f(x)f +

71=1

Σ α

Ύη

πι

(χ)φ

(χ) - 2α

/(χ)φ

(χ)

Vra=l

(2.39)

Now, substituting for

lf(x)-A

(x)}

in Equation (2.36) using Equation (2.39) gives

71=1

yZ α

τη

πι

(χ)φ

(χ) I - 2α

/(χ)φ

(χ)

771=1

¿¿x

(/(*))

•

\n=l

ll/(*)ll

<ix 4-

J^ α

πι

{χ)φ

{χ) - 2α

/(χ)φ

(χ)

Vra=l

71=1

n/o*

η=1

Λ6 / JV

/ Σ

^αη^τη(^)^η(^) \ dx - 2α

/(χ)φ

(χ)άα

iV .6

_ _ /»o /»o

^ / α

πι

(χ)φ

(χ)άχ - 2a

¡(χ)φ

{χ)άχ

m=l^

Because the functions </>

(x) are orthonormal it follows:

y^ / α

πι

Ύη

(χ)φ

(χ)άχ = (a

)(a

m=l^

)=ocl

(2.40)

(2.41)

44 FOURIER SERIES

By the definition of the Fourier coefficients c

it follows that

ί{χ)φ

x)dx

—

(2.42)

Substituting the results of Equations (2.41) and (2.42) into Equation (2.40) gives

E =

\\f(x)\\

+ J2«-

nCn)

n=l

\\f(x)\\

Y^(a

-2a

+ c

-c

)

n=l

= ¡i/(*)ii

E[(

«-

«)

n=l

(2.43)

n=l n=l

The mean-square error E, as given in Equation (2.43), is made up of three terms:

the norm squared of the function /, the sum of the square of the differences of the

arbitrary coefficients a

and the Fourier coefficients c

, and the sum of the squares

of the Fourier coefficients c

of / based on the orthonormal functions φ

(χ)- For a

given / and orthonormal set φ

(χ), the first and third terms on the right-hand side

of Equation (2.43) are fixed. Consequently, the mean-square error E is minimized

when a

— c

. That is, the Fourier coefficients c

represent the optimal set of

coefficients in the finite sum representation of /. This result is formalized in the

following theorem.

Theorem 2.3 (Best Approximation) Given a function f and an orthonormal set of

functions {φ

(χ)} from C

(a,

b),

the mean-square error

defined by

) - ^θί

{χ)

n=l

is minimized when a

= c

(n =

1,2,...

,iV), where each c

is the Fourier

coefficient given by

Cn = / /(χ)φ

(χ)άθί

In this case, E is given by

E=\\f(x)\\

~J2

(2.44)

BESSEL'S INEQUALITY 45

2.9 BESSEL'S INEQUALITY

A direct result of the material presented in Section

2.8 is

known

BessePs inequality.

Using Equation (2.44) we may write

f(x) - Σα

(χ)

which gives

n=l

ii/(*)ii

-E

)

n=l

n<ll/W

(2.46)

n=l

The expression in Equation (2.46) is Bessel's inequality.

Recall that / is a member of C

(a, 6), so that we know both / and f

are integrable

functions. That is,

\\f(x)\\

= / f(x)f(x)dx = M (2.47)

where M is some finite, fixed number. Combining Equations (2.46) and (2.47) yields

Σ^<Μ

(2.48)

n=l

Because the inequality (2.48) holds for any positive integer N, it has strong implica-

tions by providing an upper bound for a nondecreasing sequence of partial sums. As

a result, the sequence of partial sums

n=l

N =

1,2,3,...

converges. Which means the infinite series made from the squares of the Fourier

coefficients

converges. Because this series converges, the nth term of the series

must go to

zero.

It follows from this that

lim c

—

The preceding arguments establish the validity of the following theorem.

Theorem 2.4 Let c

(n = 1,2,3, ...) be the Fourier coefficients corresponding to

a given function f and an orthonormal set of function {φ

(χ)} from C

(o,

b).

follows that

lim c

= 0

46 FOURIER SERIES

This general result may be applied to a specific case, such as the set of orthogonal

functions

) =

sin

(T)

on the interval (0, c). As shown in Section

2.5,

if / is

function belonging to C

(0, c)

then the Fourier sine series corresponding to / is

n=l

with

/(*)~X>ny-sin(^)

K =

j f{x)y¡\^^-)dx

Applying the result of Theorem 2.4 gives

lim b

= 0

n—»oo

It is left as an exercise (Exercise 2.29) to show the Fourier cosine coefficients a

(n = 0,1,2,...) for the orthonormal functions

(χ) = "7= Φη{ζ) = \ -

COS

( )

on the interval

[0,

have limit zero as n goes to infinity.

2.10 PIECEWISE SMOOTH FUNCTIONS

The next objective of this chapter is to establish sufficient criteria on / from C

(a, b)

for convergence of its Fourier series correspondence to /. To that end, a subspace of

(a, b), the set of piecewise smooth functions, will be introduced.

Suppose / is an element of C

(a, 6), with discontinuities given by the set

{#i,

#2, · · ·, Xn}- Let Xk be an arbitrary number in (a,

such that the right-hand

limit of / at Xk, denoted by /(xfc+), exists. The right-hand derivative of / at

defined by

(

)

= lirn

fc+)

(2.49)

provided the limit exists. The left-hand derivative of / at Xk is defined in a similar

way. That is, if f(xk~) exists, then

{xk)

= lim

f(x)

(

-î

(2.50)

x—+x,

X %k

It is left as an exercise (Exercise 2.31) to show that if / is differentiate at Xk in its

domain, then both one-sided derivatives of / exist and are equal at

Xk-

PIECEWISE SMOOTH FUNCTIONS 47

EXAMPLE 2.5

Determine if the one-sided derivatives of / defined by

2x - 1 x < 1

f(\—( 2x

—

1 x < 1

exist at x = 1. Find the value of the one-sided derivative in the event it does

exist.

Right-hand case: Note that /(1+) = 0, so that it is possible for the right-

hand derivative to exist at x

1. Using Equation (2.49)

(., =

Ä^±I

lim

X-+1+

X — 1

= lim

i+ y/x-1

= oo

Consequently, the right-hand derivative of / does not exist at x = 1.

Left-hand case: Here, /(l—) = 1, so it is possible for the left-hand deriva-

tive of / at x

= 1

to exist. Using Equation (2.50)

/(*)-/(!-)

2x-

x -

2(x

x-1

-1-1

-1

-2

■1

-i)

/L(1)

= um

x—>1~

= lim

x-^l-

= lim

x—>1~

= lim

x-^l- X — 1

= 2

so the left-hand derivative of / at x

= 1

is 2.

The Example 2.5 implies that the function /' may fail to be piecewise continuous

even when the function / is piecewise continous.

A piecewise continuous function on the interval (a,

is said to be piecewise

smooth on an interval (a,

if the derivative /' of / is piecewise continuous on

(a, b). This subset of C

(a, b)is denoted by C^(a, b), and is a subspace of C

(a, 6)

because it is, after all, a collection of piecewise continuous functions,

The next theorem relates the one-sided limits of the derivative of /, should they

exist, and the one-sided derivatives of/, should they exist. That is to say, under what

circumstances are these quantities equal, provided both quantities exist.

48 FOURIER SERIES

Theorem

2.5 If f

is piecewise smooth of the interval

(a,

6),

then

every

in the

closed interval

[a,

the

one-sided derivatives

off

(from

the

right only

at a and

left

only

exist

and

are

equal

to the

corresponding one-side limits of

Proof:

Because

/ is

piecewise smooth

on (a,

know that

/' is

piecewise

continuous

on (a,

b).

Let

(x¿,

Xi+i)

an arbitrary representative of the finite number

of subintervals

of (a,

which

/' is

continuous

and

both

f'(xi+)

and

f'(xi+i

— )

exist. Arguments will

presented

show that

the

theorem's conclusions hold

for

the interval (x¿, Xi+i). Because these arguments are valid

for

any of the finite number

of subintervals

of (a,

b),

the

result

may be

extended

to the

entire interval

(a, b).

Let

be an

interior point

(x¿,

x¿+i). Therefore

f'(xk)

exists, and by the result

of Exercise

2.31,

follows that

{xk)

= f(xk) =

{xk)

Because

/' is

continuous

at x

have

\im

f(x)

f'(x

)

x—+x

and

limJ'(x)

f'(x

)

x—*x

Now, consider

the

right-handed derivative

of / at x¿. The

value

of f(xi) is

involved

determining

the

limit

of the

difference quotient. Because

/ is, itself,

piecewise continuous

on (a,

b),

we may

represent

f(xi) as

/(x¿+). Doing

means

continuous

on [x^x] and

differentiable

(x¿,

x),

where

x is an

arbitrary point

such that Xi

< x <

x¿+i,

shown

Figure 2.12.

Now,

{Xi)

= km ——

(2.51)

χ-+χΤ

X Xi

-\—i

1 h-

C X

Xi+i

Figure 2.12 The relative positions

andxi+i.

Using

the

continuity

of / on

[xi,-x]

and the

differentiability

of / on

(x¿,#),

the

Mean Value theorem

for

derivatives

may be

applied

assert

the

existence

point

(xi,x) suchthat

f(x)

- f(Xj)

Substituting

for the

difference quotient using Equation (2.51) gives

(xi)=

lim /'(c)

X—*X~¡

PIECEWISE SMOOTH FUNCTIONS 49

As right-hand endpoint x approaches xi from the right, it follows that the number c

will approach xi as well because Xi < c < x. Since the right-hand limit of /' exists

at Xi it follows

(xi)=

lim /'(c) =/'(*«+)

Similar arguments establish the result for the right endpoint. That is,

i+1

)

= Jim f'(c) =

f'(x

i+1

Theorem 2.5 states that the piecewise continuity of /' is sufficient to guarantee

the existence of the one-sided derivatives of /. The converse in not true, as shown in

the next example.

■ EXAMPLE 2.6

Suppose the function / is defined as

-A

»)=if-i :t°

The right-hand derivative of / at x = 0 is given by

, , , x

cos è

—

JRV*) —

0 < x cos

x^0+

—

9 1

cos^

lim

x^0+

lim x cos

—

x-*0+ X

< x for x Φ 0

Because

it follows

0 < lim x cos

—

< lim x

—

X-+0+

X x^0+

so that by the squeeze theorem we may conclude

lim x cos

—

= 0

x->0+

Similar analysis for the left-hand derivative of / at x

0 applies giving

(0)=0

and /¿(0)=0

Next, the one-sided limits of /

;

at x

0 are considered. The formula for /'

f'(x)

—

cos

—h sin

—

x x

50 FOURIER SERIES

Evaluating the limit of /' as x approaches 0 from the right gives

lim

•-♦0+

1 1"

cos

—h sin

—

X X

= 0 + lim sin

—

x-+0 x

Because the remaining limit on the right-hand side does not exist, the right-

hand limit of /' does not exist at x = 0. A similar result occurs for the left-hand

limit.

2.11 FOURIER SERIES CONVERGENCE

This section presents a theorem on the convergence of Fourier series. Prior to

stating and proving the theorem, three preliminary results, that are eventually used

to establish the theorem, are presented. The first involves an alternative form of the

Fourier series. Both of the other two concern lemmas. For sake of convenience, the

interval on which the Fourier series will be considered in this section is specifically

(—7Γ,

7Γ)

instead of the general interval (—c, c). Once the Fourier convergence result

is establish for (—π, π), it will be generalized to the general interval (—c, c) in Section

2.12.

2.11.1 Alternate Form

The general Fourier correspondence for a function / from C

(—c, c) is

f(x) ~ — + 2^ [

COS

[ J +

n Sin (^ J j

n—1

For c =

7Γ,

this expression reduces to

f(x)

rsj

— 4- V^ (a

cos

nx + b

sin nx)

where

n=l

and

1 Λ

ln = - /(·

x) cos nxc/x

x) sin nxcfa

n = 0,1,2,3,..

n =

l,2,3,...

(2.52)

(2.53)

(2.54)

are the Fourier coefficients. Substituting formulas (2.53) and (2.54) into the corre-

spondence given in Equation (2.52) gives

1 Γ

i _°°_ Γ Γ

f(x) ~ — / f(s)ds-\— 7 cosnx / f(s)cosnsds

FOURIER SERIES CONVERGENCE 51

+ sinnx / f(s)sinnsds\

1 Γ^ 1 _°°_ Γ Γ^

~ —- / f(s)ds-\— /| / f (s) cosnscosnx ds

+ / /(s)sinnssinnx ds (2.55)

Note: The terms cosnx and sinnx may be moved inside the definite integrals

because the variable of integration is ' V and not "x" in Correspondence (2.55). The

trigonometric identity

cos(a

—

β) = cos a cos β 4- sin a sin β

with a = ns and β = nx is used to transform Equation (2.55) to

2π7_

which is the desired alternate form of the Fourier series corresponding to / on the

interval (—π,π).

2.11.2 Riemann-Lebesgue Lemma

The lemma presented in this section is actually a special case of a more general

statement of the Riemann-Lebesgue Lemma.

Lemma 2.1 (Riemann-Lebesgue) If the function g(u) is piecewise continuous on

the interval

[0,

π],

then

lim Γ

g(u)sm(

{2n

+ 1)u

)du = 0 (2.57)

n-+ooj

\ 2 )

where n

represents

positive integers.

Proof:

Using the trigonometric identity

sin(a + ß) = sin a cos ß 4- cos a sin ß

with a

—

u/2 and ß = nu the integral in Equation (2.57) may be evaluated by

r ({2n + l)u\

J g(u) sin

I I

= / g(u) sin (u/2 + nu) du

= / [#(w) sin(u/2) cos nu + #(u) cos(u/2) sin nu] du

= |ûn + |δη (2.58)