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

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52 FOURIER SERIES

where a

are the Fourier cosine coefficients of the function g(u) sin(u/2) on the

interval (0, π), and b

are the Fourier sine coefficients of g(u)sm(u/2) on the

interval (0, π). Taking the limit of both sides of Equation (2.58) and applying results

from Section 2.9 and Exercise 2.29, that is

lim a

= 0

and

lim b

—

to Equation (2.58) gives

Γ .

. /(2n + l>\ , i. (* 7Γ, \

Λ π

km / g(u) sin du = lim -a

+ -b

)

= 0 U

n—>oo JQ \ 2 J n^oo \ ¿ Z /

2.11.3

Dirichlet Kernel Lemma

The Dirichlet kernel is defined as

(2.59)

71=1

where N is any positive integer. Before presenting the lemma related to the Dirichlet

kernel, two properties of DN(U) will be given.

The first involves the definite integral of DN(U) on the interval

[0,

π].

That is,

cosmx

/»7Γ /»7Γ -1 -'

/ D

(u)du = / - + Σ<

Jo J

o I

„=i

= / -du+ I I > cos nul du

2 Jo \h )

eos nudu

n=l

The second property related to the Dirichlet kernel is

sin[(2AT + lW2]

JUN{U)

(2.60)

(2.61)

2sin(i¿/2)

It is left as an exercise (see Exercise 2.18) to establish the validity of Equation (2.61).

Lemma 2.2 Suppose that a function g(u) is piecewise continuous on the interval

[0,

π],

and the right-hand derivative g'

(0) exists. Then

Γ π

Jim / g(u)D

(u)du = -#(0+)

(2.62)

FOURIER SERIES CONVERGENCE 53

where

DJSÍ{U)

is the Dirichlet kernel

DN(U)

= 1 + J2

cos

n=l

Proof:

Begin by writing the integral in Equation (2.62) as

g(u)DN(u)du =

[ [g(u) - 0(0+) + ^(0+)] D

(u)du

/»7Γ /»7Γ

= / \g(u)-g(0+)]D

(u)du+ g(0+)D

(u)du (2.63)

Jo Jo

Each of the integrals on the right-hand side of Equation (2.63) will be considered

individually. First, using the the second property of the Dirichlet kernel given in

Equation (2.61)

[g{u) - g(0+)} D

(u)du =

Γ . . . ,

.. sin U2N + l)«/2] ,

Jo 2sin(u/2)

(2.64)

Γ 9(u) - 9(0+)

2sin(w/2)

Γ g{u) -

(0+) u/2

2{u/2) sin(«/2)

(u)-g(0+) u/2

sin[(27V + l)i//2] du

(2ΛΓ + l)u

—

0 sin(u/2)

sin

(27V + l)u

Let

h{u) =

g(u) - 0(0+) u/2

(2.65)

(2.66)

w —

0 sin(u/2)

the first two terms of Equation (2.65). For the sake of argument, it must be established

that h is piecewise continuous on (0,7r).The primary concern to this end is the fact

that h is undefined at u

0. For all u in (0, π) the terms in the product that defines h

are piecewise continuous, assuring the piecewise continuity of h there as well. The

piecewise continuity of h hinges on the right-hand limit of h at u = 0.

lim h(u) — lim

u->0+ u->0+

lim

g(u)-g(0+)\

( u/2

u-0 ) \sm{u/2)

g(u)-g(0+)\

u-0 )

u/2

lim . . . . ,

u->o+

\sm(u/2)

, (2.67)

provided the individual limits of the right-hand side of Equation (2.67) exist. The

first limit is just that for 0^(0), which exists by hypothesis, and the second limit is a

54 FOURIER SERIES

version of the familiar sin x over x limit from calculus I. Therefore, h is piecewise

continuous on (0, π).

The piecewise continuity of h allows the application of Lemma

2.1,

so that

(27V + l)u

/»7Γ /»7Γ

lim / [g(u)

—

#(()+)] Djy(u)du — lim / h(u) sin

N->oo J

ÍV-+00 JQ

= 0 (2.68)

As for the second integral on the right-hand side of Equation (2.63), it follows that

Jim / g(0+)D

(u)du = lim 0(0+)-

= ff(0+)| (2.69)

Combining the results given in Equations (2.68) and (2.69), it follows that

lim / g(u)DN(u)du

—

lim / [g(u) - g(0+)]D

(u)du+ lim / g(0+)D

(u)du

+ ^(o+)

= ^(0+) D (2.70)

2.11.4 A Fourier Theorem

It is now possible to state and prove a theorem concerning the convergence of a

Fourier series.

Theorem 2.6 If function f is

(i) piecewise continuous on (—π, π), and

(ii) 2π-periodic for all x such that — oo < x < oo

ί/îen i/îe Fourier series

— + /_^ (a

cos

nx + b

sin nx)

n=l

with

1 /*

n = 0,l,2,...

1 Γ

= — / /(s) cos(ns) ds

1 /*

bn = — f(s) sin(ns) ds

^ J-7T

and

1 Γ

n= 1,2,3,...

FOURIER SERIES CONVERGENCE 55

converges to

/(s+)+/(x-)

for all x where both one-sided derivatives

(x)

and

(x)

exist.

A couple of remarks before the proof of this statement

presented. First, piecewise

continuity and the existence of one-sided derivatives are sufficient conditions for

convergence. Second, if the function / is continuous at x, it follows that f(x+) —

jf (x) =

f(x-),

so that the Fourier series converges to f(x).

Proof:

The piecewise continuity of / means the Fourier coefficients a

and b

exist for all appropriate values of n. The corresponding Fourier series for /, in

alternative form as given in Equation (2.56), is

1 Γ

1 _°°_ Γ^

— / f(s)ds + -^T f(s) cos n(s-x)ds (2.71)

The Nth partial sum S^ of this series is

(x)=.—

f f(

)ds + -y* [ f(s)cosn(s-x)ds (2.72)

The Dirichlet kernel in terms of s

—

x is

—

x) = - 4-

2_.

cos

n(s

—

n=l

so that the partial sum in Equation (2.72) may be written as

1 Γ

(x) = - \

f{s)D

(s - x) ds (2.73)

The objective is to establish the desired convergence result for any real x, not just

those in the interval (—π, π). By the 27r-periodicity of / and the Dirichlet kernel,

integration of the integrand in Equation (2.73) over any interval of length 2π will

result in the same number. Therefore,

(x) = - [ f(s)D

(s - x)ds (2.74)

Jx-π

where x is any real number, and is the midpoint of the arbitrary 27r-long interval. The

integral in Equation (2.74) will be split into the two following integrals for the sake

of analysis.

(x) = - i Í f(s)D

(s - x)ds + / f{s)D

(s - x)d

{.Jx—TV

(2.75)

56 FOURIER SERIES

Each integral on the right-hand side of Equation (2.75) will be simplified using

Lemma 2.2, after making an appropriate change of

variable.

For the first integral, the

change of variable will be u = -s + xso that

f(s)DN(s

—

x)ds = — I f(x

—

U)DN(U)CIU

J X — 7Γ J TV

f(x-u)D

(u)du (2.76)

Next, let g(u) = f(x

—

u) in Equation (2.76). To apply Lemma 2.2, it must be

that g{u) is piecewise continuous on (Ο,π), and g'

(0) exists. Piecewise continuity

follows from the piecewise continuity of /. To establish the existence of the right-

hand derivative of g at x — 0, begin with the definition of the right-hand derivative.

That is,

where

g(0+) = limj(u)

u—>0+

= lim f (x

—

= lim f(y) (y = x-u)

y-*x~

= f(x-) (2.78)

Substituting for g(u) and 5(0+) in Equation (2.77) in terms of / gives

(0)

= lim /(* - ") - /(*-)

yñV ;

„^0+ w-0

= - lim

f{y)

f{x

]

(y = x-u)

V-+X- y - x

= -fí(x)

(2.79)

Therefore, g

(0) exists whenever

(x),

which is, by hypothesis, for real x. Conse-

quently, we may write

lim / f(s)D]y(s

—

x)ds = lim / g{u)D]sr(u)du

= >+)

-\fix-)

(2.80)

for all real x.

FOURIER SERIES CONVERGENCE 57

The second integral on the right-hand side of Equation (2.75) is analyzed in a

similar way. In this case, the change of variable is u

—

x. Then,

ρχ+π ρπ

¡ f{s)D]sf{s

—

x)ds = / f(u + x)D]y(u)du

Jx JO

g(u)D

(u)du (2.81)

where g(u) = f(u + x). As with the first integral, in order to apply Lemma 2.2, we

must first show that the right-hand derivative of g exists at x=0. To that end,

¿(0) =

M*tiM

u-^0+ U — 0

/(" + *)-/(*+)

u-»0+ U — 0

,. f(y) - /(*+) , ^ ,

= lim (y = u + x)

y->x+ y-x

{x)

(2.82)

Note that, as before, g is piecewise continuous by virtue of the piecewise continuity

of/, and #(0+) = /(#+) because,

0(0+) = limo(u)

u->0+

= lim f(u + x)

u-+0+

= lim f(y) (y = u + x)

y-+x+

= f(x+) (2.83)

Then, by Lemma 2.2 we may right

rx+π ρπ

lim / f(s)D]sf(s

—

x)ds = lim / g{u)DN(u)du

N->oo

= |.(o

)

= |/(x+) (2-84)

Using Equations (2.80) and (2.84) in the process of taking the limit as

goes to

infinity of both sides of Equation (2.75), we have

lim SN(X) = lim — / f(s)D

—

x)ds + / f(s)D

—

x)ds

N^oo

N-^oo π |_Λ-

[In*-) +

£/(*+)]

/(*-)+/(*+)

(2.85)

58 FOURIER SERIES

That

is,

the sequence of partial sums of the Fourier series corresponding to / converges

to the average of the one-sided limits of / wherever the one-sided derivatives of /

exist. D

In many applications considered in latter portions of this text, a given function /,

serving as a boundary or initial condition, will be piecewise smooth on the interval

(—7Γ,

π). Such a function may be extended to a 27r-period function F defined on

the entire real number line. The function F is piecewise smooth over the entire real

number line because it inherits this characteristics from /.

A piecewise smooth function is such that both one-sided derivatives exist for

each x, as stated in Theorem 2.5. Consequently, the 27r-periodic extension F of

the piecewise smooth function / is such that the Fourier series of / on the interval

(—7Γ,

π) converges to

F(x-) + F(x+)

for all real x. This result is formalized as a corollary to Theorem 2.6.

Corollary 2.1 Suppose function f is piecewise smooth on the interval (—π, π). Let

F represent the 2π-periodic extension of f to the entire real number

line.

Then, the

Fourier

series given by

— + 2_\ [

cos

nx -f b

sin nx]

n=l

where

and

1 Γ

— — I f(x)cosnxdx

1 Γ

—

/ f(x)sinnxdx

π J-TT

n = 0,1,2,...

—

1,2,3,...

converges to

F(x-)+F(x+)

for all real numbers x.

2.12 2C-PERIODIC FUNCTIONS

The objective of this section is to generalize the convergence result presented in

Corollary 2.1 from a piecewise smooth function that is 27r-periodic on the real

numbers to one that is 2c-periodic, where c is any positive real number.

Suppose the function / is piecewise smooth and 2c-period on the real number

line.

Further, suppose at each discontinuity x¿

) = n**-)+f(**+)

2C-PERI0DIC FUNCTIONS 59

Define function g to be

g(s) = f (") (2.86)

Defining g in this way is essentially making a change of variable x = — in the

function /. Defining g in this way results in the following facts:

• The function g is piecewise smooth on the real number line because / is

piecewise smooth, and g is simply a composition of /.

• The function g is 27r-periodic. This follows by

/■ « N , /φ + 2π)

g(s + 2n) = f

1 K }

)

9(s)

and the 2c-periodicity of /.

• The function g at any number s is the average of the one-sided limits of

at 5.

This follows by

9is) = /(f)

g(s-)+g(s+)

Because g is piecewise smooth and 2π-periodic on the entire real number line,

Corollary 2.1 may be applied to assert that the Fourier series

where

and

— + 2_, \

cos ns

+ b

sin

ns]

n=l

1 Γ

—

/ g(s) cos nsds

π J-7T

1 Γ

K = - / g(s)

n = 0,l,2,...

converges to

sinnsds n

—

1,2,3,...

60 FOURIER SERIES

for all real numbers s. Making the (reverse) change of variable s = ^f, the expres-

sions above become

Σ,

cos

V~JJ

n=l

where

1 /*

/πχ\ /ηπχ\ π

—

/ g I—J cos

(

)

—

αχ η = 0,1,2,...

η = 1,2,3,.

and

_ 1 í

/πχ\ . /ηπχ\ π ,

π/_

ητ)

8ΐη

Ι—Jc

the series converges to

0(ΊΓ-)+0ΟΓ+)

And, the relationship between x and 5 is such that

so that, in fact, we have

«n = - / f(x)

COS

( J dx

and

= - f(x) sin f J dx

the series converges to

/(s-)+/(s+)

for all real x.

These preceding arguments provide the necessary support for the following theo-

rem.

Theorem 2.7 Suppose function f ispiecewise

smooth,

2c-periodic, and

defined

such

that

/or a//

retf/

numbers x. The Fourier series representation

/(x) = — + 2^ [

η cos ^ J + b

sm ^—■ J J

n=l

n = 0,l,2,...

n = 1,2,3,...

CONCLUDING REMARKS 61

with

and

1 f

/TL7TX\

= - I f(x)cos[ ) dx n

—

0,1,2,.

= - / f(x) sin ( j dx

\ c )

n= 1,2,3,...

is valid for all real numbers x.

2.13 CONCLUDING REMARKS

Recall that a Fourier series corresponding to a function / is said to be valid on an

interval (a,

if the series converges to / at all x except, perhaps, a finite number

of locations. Theorem 2.7 establishes the validity of the Fourier series, made up

of both sine and cosine terms, for 2c-periodic, piecewise-smooth functions on the

entire real number line. For many applications presented in this text, the domain

on which the representation is desired is finite, such as

[0,c].

The imposition of

boundary conditions in these instances frequently result in sine-only or cosine-only

Fourier series. The implications taken from Theorem 2.7 are sufficient to establish

the validity of the such series in these instances.

To illustrate the intent of

the

previous paragraph, suppose we want to establish the

validity of the series representation

f(x) = 2_\ b

sin(mrx/c)

n=l

Suppose / is piecewise smooth on [0,c] and

/(χ)=

/(χ+)+ /(*-)

for all x in (0,c). Let F be the function constructed by first doing an odd extension

of / to the interval [—c,

c],

and then extending that result to a 2operiodic extension

to the entire real number line. Function F is piecewise smooth on the real number

line.

Further, it follows that

Fix) =

^+Η^*-)

for all x except, perhaps, those x for which x = kc, where k is any integer. Under

these conditions, we know by Theorem 2.7 that the Fourier series

αο ν^ Γ /ηπχ\ . . /ηπχλΐ

+ ^[a„cos(—J+6

sin(—jj

converges to F for all x except, perhaps, those x given by kc. Note this validity

includes all x in the interval (0, c). Because F is an odd extension, and therefore an