Lecture 3: Fourier Series and Fourier Transforms

Key pointsA function can be expanded in a series of basis functions like

,

where are expansion coefficienct.

When are trigonometric functions, we call this expansion Fourier expansion.

Fourier Series: For a function of a finite support ,

.

where and ,

or .

Fourier Series: For a periodic function ,

Fourier Transform: For a function ,

Forward Fourier transform:

Inverse Fourier transform: .

Maple commandsintinttransfourierinvfourieranimate

1. Fourier series of functions with finite support/periodic functions

If a function is defined in or periodic as in , it canbe expanded in a Fourier series:

where . The Fourier coefficients are determined by the following integrals:

and , or .

Example

= = 5

= = n4

= = n3

.

Visual inspection

We calculate the Fourier coefficients up to n=10

(1)(1)

(3)(3)

(2)(2)

Compare the original function and the truncated Fourier series.

Truncated Fourier series vs original function Error due to the truncation

Exercise 3.1

Expand in a Fourier series.

Answer

If the support is

Functions defined in a finite region: . Introducing a new variable

, we have a new function defined in as

.

Exercise 3.2

Transform defined in to an equivalent function defined in .

Answer

If the period is LIf a function has a period : , use a new variable

. Then, the function can be always expressed as

Common sense

When is defined in or periodic as its Fourier series is given by

.

where the Fourier coefficients are given by

and .

You should prove this.

Basis functions

A function can be expanded using a set of orthonormal basis functions :

where satisfies the orthonormal condition

.

The upper and lower bounds, and , can be . Fourier expansion is an example. We will discuss general cases later in the Linear Algebra section.

Common senseKronecker delta

.

Introducing normalized basis functions: , ,

( ), Fourier expansion can be expressed as

.

The basis functions satisfy orthonormality. It is straightforward to prove it using the following integrals:

= for .

= 0 for .

= for .

= 0 for .

.

Using the orthonormal relation, we can easily find the coefficients:

.

.

.

Alternatively, we can use as a basis function which is orthonormal:

.

Then, Fourier series is expressed as

.

and the Fourier coefficients are given by

2. Fourier transforms

Fourier series tries to represent a function using a descrete set of waves. That is possible when the function has a finite support or has a finite period. For other cases, a continuous set of waves are needed.

.

Function is called Fourier transforms of . can be obtained by the Fourier transforms:

. [In general, is a complex function even when is real.]

MapleMaple package inttrans includes Fourier transformation.You need to load the package before ussing Fourier transformation commands.

= 2

,

=

Example

Original function:

Fourier transform: 0

Inverse Fourier transform: = e

which is the same as the original function.

Using Maple functions,

0

= e

Fourier transform of derivatives

Consider a function which vanishes as . Then, the Fourier transform of the derivative of is given by

.

Differentiation in x space = multiplecation of -i k in k space.

Fourier transform of linear ODE'sSuppose that satisfies a linear ODE

.

Then, its Fourier transform satisfies.

This means for almost all exept for the roots of .We will discuss this method in ODE section.

Fourier integral theorem and the Dirac's delta function

Fourier integral theorem states .

This theorem provides the foundation of Fourier transform.

Rigourously speaking, the order of integrals in the Foureir integral theorem cannot be swapped. Ignoring this mathematical rule, we write the Fourier integral theorem as

.

As the mathematical rule told us,the integral in the parentheses does not converge. Nevertheless, Dirac called it delta function

,

and the Fourier integral theorem is wirtten as

.

x) is zero everywhere except for x=0 and diverges atx=0. In adition, it satisfies

,

(4)(4)

(5)(5)

Forward vs. inverse transformation

Difference between forward and inverse fourier transformation is mainly the factor in the

inverse transformation. However, that is not only the choice. We need to satisfy the Fourier

integral theorem but it does not say where we should put . The following symmetric

definition of Fourier transform is also commonly used.

,

.

Parseval's theorem

.

Exercise 3.3

Confirm the Parseval's theorem for .

Answer

3. Examples in Physics

1. Power spectrum

decays as

.

Find its power spectrum where is Fourier transform of .

Explicit integration gives the same result.

(6)(6)

(7)(7)

(9)(9)

(8)(8)

(10)(10)

The power spectrum of is given by

a C bI form

= simplify

C2

factor Omega^2-2*Omega*omegaC omega^2

C2

= = 1

Homework: Due 9/11, 11am

3.1 ParityFor a real function , show that

if the function is even, , then . and are all real.

if the function is odd, , then , and are pure imaginary.

3.2 Average

Consider a function .

The average value of the function is defined by .

Show that .

3.3 Piecewise constant functionConsider a periodic function

and

Show that its Fourier series is give by

.

3.4 Change of interval Consider a function . Show that it can be expanded in the following way:

,

where and .

3.5 Real functionsIf is a real function, show that .

3.6 Fourier transform of derivativesAssuming that (with a sufficienctly strong convergence), prove the derivative

formula .

[Use the method of integral-by-part.]