Fourier Series Methods† National Chiao Tung University Chun-Jen Tsai 12/9/2019 † Chapter 11.1 ~ 11.3 in the textbook.

Fourier Series Methods

^†

National Chiao Tung University Chun-Jen Tsai 12/9/2019

† Chapter 11.1 ~ 11.3 in the textbook.

Orthogonal Functions & Inner Product

 Vectors in linear algebra are not just n-tuples

 If u and v are two n-tuple vectors in 3D-space, then the inner product (u, v) possesses the following properties:

 (u, v) = (v, u) (inner product is commutable)

 (ku, v) = k(u, v) (k is a scalar)

 (u, u) = 0, if u = 0 and (u, u) > 0, if u  0

 (u+v, w) = (u, w) + (v, w) (inner product is distributable)

 The definite integral of two functions, f₁ and f₂, over an interval [a, b] possesses the same properties as well  we can define “inner product” for functions

Inner Product of Functions

 The inner product of two functions f₁ and f₂ on an interval [a, b] is the number

 Two vectors are “orthogonal” if the inner product is zero

 function inner product should be defined similarly:

two functions f₁ and f₂ are orthogonal on an interval [a, b] if



 ^b

a f x f x dx

f

f , ) ( ) ( )

( _{1 2 1 2}

0 )

( ) ( )

,

( f₁ f₂ 



a^b f₁ x f₂ x dx 

Orthonormal Set of Functions

 A set of real-valued functions {

f

₀,

f

₁,

f

₂, …} is said to be orthogonal on an interval [a, b] if

 The norm of a function

f

is defined as ||

f

|| = (

f

,

f

)^1/2. That is,

If {

f

_n} is an orthogonal set on [a, b] and ||

f

_n|| = 1, n, then {

f

_n} is an orthonormal set of functions on [a, b].

. ,

0 )

( ) ( )

,

( ^b x x dx m n

a m n

n

m

^f



 ^{f f}

 

f

. ) ( )

(x ^



a^b

f

² x dx

f

Orthogonal Series Expansion

 If {

f

_n(x)} is an infinite orthogonal set of functions on the interval [a, b], is it possible to determine a set of

coefficients c_n, n = 0, 1, 2, … such that

f(x) = c₀

f

₀(x) + c₁

f

₁(x) +  + c_n

f

_n(x) +  ? To find the coefficient of

f

_n, we compute (f ,

f

_n)

(f ,

f

_n) = c₀(

f

₀,

f

_n) + c₁(

f

₁,

f

_n) +  + c_n(

f

_n,

f

_n) +  Since {

f

_n(x)} is an orthogonal set, (

f

_m,

f

_n) = 0,  m  n.

Therefore,

2 and ) , (

n n n

c f

f



f

( , ) ( .)

) (

0 2



^





n

n n

n x

x f

f

f

f

f

Completeness of an Orthogonal Set

 In previous discussion, we have

if f(x) can be represented as a linear

combination of

f

₀(x) ~

f

_(x) in the vector space S.

However, not every functions in S can be represented as a linear combinations of the functions in {

f

_n(x)}.

This is true only when {

f

_n(x)} is a complete set of S, i.e., when {

f

_n(x)} is a vector basis of S.

, ) ) (

, ) (

(

0 2



^





n

n n

n x

x f

f f

f f

Periodic External Forces

 Recall that a linear 2^nd-order DE:

where f(t) stands for the external force imposed on the (undamped) system. Often, f(t) is a periodic function (over an interval of interest).

 Question: Is there a systematic way to represent a general periodic function?

 Well, Taylor series may work, but can we do better?

),

2 (

2 0 2

t f dt x

x

d 





Properties of a Periodic Function

 Definition: The function f(t) defined for all t is said to be periodic provided that there exists a positive number p such that f(t + p) = f(t) for all t. If p is the smallest

number with this property, then p is called the period of the function f.

 Remarks:

 Linear combinations of two (or more) periodic functions will still be a periodic function.

 If we use a set of periodic functions as basis functions to represent other periodic functions, they should work better than if we use {1, x, x², x³, …}, as in Taylor series.

Selection of Periodic Basis

 In 1822, J. Fourier asserted that every function f(t) with period 2



can be represented as a linear combination of sin nt and cos nt, as follows:

 Really? How about the function:



^{cos sin}



^,

2 ₁

0



^







n

n

n nt b nt

a a

f

t

 1

- –1

2 3

Fourier Series

 Note that the set of trigonometric functions:

{1, cos t, cos 2t, cos 3t, …, sin t, sin 2t, sin3t, …}

are orthogonal on the interval [–



,



].

 The Fourier series of f(t) on [–



,



] is defined as :

where



^{cos sin}



^,

) 2 (

1

0



^









n

n

n nt b nt

a a t

f



-

 ^

  ^{f t dt}

a 1 ( )

0

. sin

) 1 (

, cos

) 1 (





-  -

 ^





 

 ^{f t ntdt b f t ntdt}

a_{n n}

),

 f(t

a_n cos nt

|| cos nt ||²

The projection vector of f(t) onto cos nt is . Thus,

nt ), cos

(t

f cos nt

|| cos nt ||²

Fourier Series with Period 2p

 Note that the set of trigonometric functions

is orthogonal on the interval [–p, p].

 The Fourier Series of a function f(x) on (–p, p) is:







  3 ,

sin 2 ,

sin , sin

, 3 ,

cos 2 ,

cos , cos

,

1 p

x p

x p

x p

x p

x p

x     





^









1

0 ( cos sin )

) 2 (

n

n

n p

x b n

p x a n

x a

f  



-

 ^p

p f x dx a 1p ( )

0





-  -

 ^p

n p p

n p dx

p x x n

p f b

p dx x x n

p f

a  

sin ) 1 (

, cos

) 1 (

Example: (1/2)

 Since p =



, we have







 -





 -







x x

x x

f , 0

0 ,

) 0 (

2 2

1

) (

1 0

) 1 (

0 2

0 0

0

 



 













 



 



 -





  -











- -

x x

dx x dx

dx x f

a ^y

 x

 -









 

 





 









2

2 0

0 0

0 0

) 1 ( 1

1 cos

cos 1

1 sin )sin

1 (

cos ) (

1 0 cos

) 1 (

n

n n n

nx n

n nxdx n

x nx

nxdx x

dx nxdx

x f a

n n

-

 -



 - -













 - 





  -













- -

Example: (2/2)

^{f (x) } _^

_

^0,_{- x,} ^-

^

_{0 }^_x^x_^

_

⁰

( Note that cos n = (–1)ⁿ )

nxdx n x

b_n 1

sin ) 1 (



0 ^{- }

 ^ 



 



^

 



 - - 





1 2 1 sin

) cos 1 ( 1 4 _n

n

n nx n nx

x

f 



Fourier Convergence Theorem

 Theorem: Let f and f  be piecewise continuous on the interval (–p, p); that is f and f  be continuous except at a finite number of points, then the Fourier series of f converges to f at a point of continuity.

At a point of discontinuity the Fourier series converges to the average:

where f(x+) and f(x–) denote the limit of f at x from the right and the left, respectively

2 ,

) (

)

(x  f x- f

Example: Converges at Discontinuity

 The following function is discontinuous at x = 0:

 The series converges to f at x  0. At x = 0, the series converges to:







 -





 -







x x

x x

f , 0

0 ,

) 0 (

2 2

0 2

) 0 ( )

0

(   f - 



 _



f

Periodic Extension

 Fourier series not only represents a function f on the interval (–p, p), but also gives the periodic extension of f outside the interval.

 When f is piecewise continuous and the right- and left- hand derivatives exist at x = –p and x = p, respectively, then the series converges to the average

[f(–p–) + f(–p+)]/2 = [f(p–) + f(–p+)]/2 at the end points:

p –p –2 p

–3 p

–4 p 2p 3p 4p x

 y

Sequence of Partial Sums (1/2)

 It is interesting to see how the sequence of partial sums {S_N(x)} of a Fourier series approximates a function. For example,

x 3

y

2 1 0

- 10 - 5 0 5 10

 

–



 –2

 –3

–4  2 3 4

x

 y

 , sin 2 cos

) 4 ( 4 , )

( ₂

1 x S x x x

S    







Sequence of Partial Sums (2/2)

x 3

y

2 1 0

- 3 - 2 - 1 0 1 2 3

x 3

y

2 1 0

- 3 - 2 - 1 0 1 2 3

x 3

y

2 1 0

- 3 - 2 - 1 0 1 2 3

x 3

y

2 1 0

- 3 - 2 - 1 0 1 2 3

(c) S₈(x) on (-, ) (a) S₃(x) on (-, )

Even and Odd Functions

 A function is said to be “even” if f(–t) = f(t) and “odd” if f(–t) = –f(t).

 Note that cos t is even while sin t is odd.

f

t t₀

f(t) = t²

–t₀ f(–t₀)

f

t f(t) = t³

f(t₀)

t₀ –t₀

f(–t₀)

f(t₀)

Properties of Even/Odd Functions

 The product of two even functions is even

 The product of two odd functions is even

 The product of an even and an odd functions is odd

 The sum (difference) of two even functions is even

 The sum (difference) of two odd functions is odd

 If f is even, then

 If f is odd, then

. ) ( 2

)

(



0



- ^

a a

a f t dt f t dt

. 0 )

( 



- a

a f t dt

Cosine and Sine Series

 If f is an even function on (–p, p), then

 Similarly, if f is odd,

. 0 sin

) 1 (

, cos

) 2 (

cos )

1 (

, ) 2 (

) 1 (

0 0 0























- - -

p n p

p p

n p

p p

p

p xdx x n

p f b

p xdx x n

p f p xdx

x n p f

a

dx x p f

dx x p f

a







. sin

) 2 (

, , 2 , 1 , 0 ,

0 ^{ }



0

 _n ^p

n xdx

p x n

p f b

n

a





Fourier Cosine and Sine Series

 Suppose that the function f(x) is piecewise continuous on the interval [0, p]. The Fourier cosine series of f is:

The Fourier sine series of f is:

Example:

 Calculate b_n as follows:













-





 - -









 x x

x x

f

0 ,

1

, , 0 ,

0

0 ,

1 )

(



^{1 ( 1)}



_.

cos 2 1 cos 1

1 1

sin ) 1 (

0 0

















nx n nx n

n

nxdx x

f b

n n

-

 -



-

 

 

 





-



- ^

. 5

5 sin 3 1

3sin sin 1

4 )

1 2

(

) 1 2

sin(

) 4 (

1



 



   

- 





^ -



x  x

n x

x x n

f

n 



1 0.5

0 x

1 0.5

0 x

S₁(x) S₂(x)

–0.5 –0.5

1 0.5 0 .

x 1 0.5

0 x

S₃(x) S₁₅(x)

–0.5 –0.5

The partial sum tends to overshoot the limiting values of f(x)  Gibbs’s Phenomenon:

Half-Range Expansions

 Sometimes, we only care about the Fourier series defined on (0, L). We can define the function f on (–L, 0) so that the expansion has a simpler form.

 Three possible choices of extension:

The third one has period L, others have period 2L.

y

x

–L L

y

–L x

L x

- L L

y

Example: f(x) = 2x – x

²

, x  (0, 2)

 We can expand f(x) to the range (–2, 2) and make it an even (f_even(x)) or an odd (f_odd(x)) function:

f_even(x) = f(–x) = 2(–x) – (–x)² = –2x – x², for x < 0, or

f_odd(x) = –f(–x) = –[2(–x) – (–x)²] = 2x + x² , for x < 0.

The Fourier expansion of f_even(x) has only cosine terms while f_odd(x) has only sine terms.

(a) Even expansion of f(x)

f_even(x)

–4 –2 2 4 x

(b) Odd expansion of f(x)

–4 –2 2 4 x

f_odd(x)

f(x) f(x)

Example: f(x) = x

²

, 0 < x < L

 Expand f(x) in a (a) cosine, (b) sine, (c) Fourier series

y

x

(a) Cosine series

- 4L - 3L - 2L - L L 2L 3L 4L

y

x

(b) Sine series

- 4L - 3L - 2L - L L 2L 3L 4L

y

- 4L - 3L - 2L - L L 2L 3L 4L x

Review: Periodic Driving Force

 When the driving force f(t) of a DE is periodic and defined over [0, p], Half-range expansion of Fourier series are quite useful. For example, the particular solution of the DE:

can be solved by first representing f(t) by a half-range sine expansion and assume a particular solution of the form:

)

2 (

2

t f dt kx

x

m d  



^





1

sin )

(

n

n

p t

p B n

t

x



Example: x"+4x = 4t, x(0)=x(1)=0 (1/2)

 Assume that 0 < t < 1 for f(t), we can use odd extension with p = 1 to get the Fourier sine series of f:

The solution x(t) should be in sine series form as well:

Note that x(t) satisfies the boundary conditions.

Substitute the solution into the DE, we have

. ) sin

1 ( 4 8

1



^ 1



- 



n

n

t n n

t 



. sin

) (



^1





n

n n t

b t

x 

. ) sin

1 ( sin 8

) 4 (

1

1

1

2

2





^



 



 -

 -

n

n

n

n n t

t n n b

n 

 



Example: x"+4x = 4t, x(0)=x(1)=0 (2/2)

 The solution of the coefficients b_n is then

The Fourier series solution can be expressed as:

which is equivalent to

in the interval (–1, 1).

). 4

(

) 1 ( 8

2 2

1



 n

b n

n

n -

-

  ^

).

1 0

( ) ,

4 (

sin )

1 ( ) 8

(

1 2 2

1  

-





^ -





n t n

t t n

x

n

n







2 sin

2 ) sin

( t

t t

x  -

-0.5-1 0 1 2

0 0.5