Lecture 3: Fourier Series and Fourier Transforms
Key points A 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
or
and
,
.
Fourier Series: For a periodic function
,
Fourier Transform: For a function
,
Forward Fourier transform: Inverse Fourier transform:
.
Maple commands int inttrans fourier invfourier animate
1. Fourier series of functions with finite support/periodic functions If a function is defined in be expanded in a Fourier series:
or periodic as
in
, it can
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)
(2)
(3)
Compare the original function and the truncated Fourier series.
Truncated Fourier series vs original function
Error due to the truncation
Exercise 3.1 Expand Answer
in a Fourier series.
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 . Answer
defined in
to an equivalent function
If the period is L If a function has a period :
defined in
, 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
where
can be expanded using a set of orthonormal basis functions
:
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 sense Kronecker 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 . [In general,
.
can be obtained by the Fourier transforms:
is a complex function even when
Maple Maple 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: which is the same as the original function. Using Maple functions, 0 =e
=e
is real.]
Fourier transform of derivatives Consider a function which vanishes as derivative of is given by
. Then, the Fourier transform of the
. Differentiation in x space = multiplecation of -i k in k space. Fourier transform of linear ODE's Suppose 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 at x=0. In adition, it satisfies ,
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
. (4)
(5) Explicit integration gives the same result.
(6) The power spectrum of
is given by
(7) a C bI form
(8)
simplify
=
C2
(9)
factor Omega^2-2*Omega*omegaC omega^2
C2
=
(10)
=
1
Homework: Due 9/11, 11am 3.1 Parity For a real function , show that if the function is even, if the function is odd,
, then , then
3.2 Average Consider a function
. and , and
.
The average value of the function is defined by
Show that
are all real. are pure imaginary.
.
.
3.3 Piecewise constant function Consider 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 functions If is a real function, show that
.
3.6 Fourier transform of derivatives Assuming that (with a sufficienctly strong convergence), prove the derivative formula [Use the method of integral-by-part.]
.
