**Project 4 Fourier
Series: a Linear Algebraic Perspective**

**I. Introduction**

The purpose of this project is to gain understanding and experience with the notion of expanding a function as a Fourier Series. This powerful method is at the heart of applications in mechanics, sound, signal processing, image processing, and many other areas of modern science and engineering. It goes hand in hand with the Fourier Transform, whose roots lie in optics.

**II. Background**

** **Review class notes on *orthogonal basis* or the textbook for** MA 2051**, Differential Equations by Professor Davis, specifically section 9.4

Recall from earlier in the course our discussions on changes of coordinate
systems. In general, finding the coordinates of a vector in a new coordinate
system amounted to straightforward solution of a linear system. We also
showed if the new coordinate system was* orthogonal*, then solving for the
new coordinates of a given vector was far easier; one could isolate each
coefficient via dot products instead of solving a system of equations. That concept lies at the heart of Fourier Series.

Our problem here is to take an arbitrary function** f(x)** defined
and finite on the for x between -1 and +1 and to expand it in terms of
the *orthogonal basis*

** {
1, cos(px), cos(2px),
cos(3px),. . . , sin( px),
sin(2px),sin(3px).
. . }**

** **

such an expansion is called a *Fourier Series of the function f. *The
coefficients, rather than being called coordinates, are called **Fourier
coefficients**, but the concept is identical.

**Maple Review**

In order to do this work, the reader will need to be able to use software to

1. Define a function

2. Do a definite integral

3. Plot a function or functions

Three brief examples should suffice for those using Maple:

**> f:= x-> x^2 + cos(2*x) +2*Pi;** defines
a function f(x)=x^{2} + cos(2x) +2p

and

**> a:= int(f(x),x=-2..2);** performs
the definite integral of f and assigns it to a

while

**> plot({f(x),g(x)}, x=0..5}; **
plots f(x) and g(x) on the same graph, for 0<x<5

__Part One__ Establishing the Orthogonality
of the Basis

**{ 1, cos(px), cos(2px),
cos(3px),. . . , sin( px),
sin(2px),sin(3px).
. . }**

** **

First we must define a "dot product" of two functions. We define it as:

so, for example,

(In most material on Fourier Series, this is called an *inner*
product instead of dot product).

Having done this, we may
work on the problem at hand, pointing out that there are really ** five**
dot products which must be shown to be 0 so that all possibilities are
covered. For example, we must show

as well as **three** others (what are they?). Use Maple to establish
all **5** results.

__Part Two__ Expanding a function in a Fourier
Series

** **

if **f(x) = a _{0} + a_{1} cos(px)
+ a_{2} cos(2px) + . . . + b_{1}sin(px)
+ b_{2} sin(2px) + b_{3} sin(3px).
.**

our problem is to find the coefficients a_{0}, a_{1},a_{2}.
. . b_{1}, b_{2},. . . This is greater aided by the orthogonality
you established in **Part One.** Just as in the case of geometric vectors,
we may take the dot product first of both sides with 1:

**f(x) · 1 = 1** ·
(**a _{0} + a_{1} cos(px) +
a_{2} cos(2px) + . . . + b_{1}sin(px)
+ b_{2} sin(2px) + . .)**

** =
1· a _{0} **by the orthogonality!

** =
2 a _{0} , **so
we can solve for

similarly, taking the dot product of both sides with **cos(npx)**
gives

**f(x)** · **cos(npx)**
= **cos(npx)** ·
**(a _{0} + a_{1} cos(px) +
a_{2} cos(2px) + . . . + b_{1}sin(px)
+ b_{2} sin(2px) + . . .)**

** =
cos(npx) ·
a _{n} cos(npx) **by the orthogonality!

** =
a _{n} **by direct integration

so **a _{n} **is easily solved for.

Finally, one may take the dot product of both sides with **sin(npx)
**and isolate **b _{n}** in a like manner. Since we have treated
n as a parameter, rather than a specific value, we have covered all cases
and have the Fourier Series of

For example, we find the Fourier Series of the function** f(x) = x ^{2}
-3x + 1. **By direct

** **

** **

** **

**so **

**x ^{2} -3x + 1 = 2/3 -(4/ p^{2})cos(px)+(4/4
p^{2})cos(2px)- (4/9
p^{2})cos(3px)...-6/psin((px)+
(6/2 p) sin(2px)-**

**(6/3 p) sin(3px)
. . .**

** **

**Exercises: **

**1. **Find the Fourier Series for each function below, whose definition for -1 < x < +1 is:

**a. f(x) = 2x+1 b.
f(x) = 7cos(2px) c.
f(x) = x ^{3} + x^{2}+ 1**

( you should have a formula for **a _{n}** and

** **

**2. **For the function in **1a** above, on the *same* graph, **plot**
it *as well as*

a. the first **two** nonzero terms of the Fourier Series for it

b. the first **three** nonzero terms of the Fourier Series for it

c. the first **four** nonzero terms of the Fourier Series for it

(so you should hand in **3** graphs altogether)

** **

3. **Prove** (on paper, not Maple) that if **f(x)** is an **even**
function (symmetric about the y axis; **f(-x) = f(x)** ) then** b _{n}
= 0** for all n

Include 3 examples of *even* functions.

4. **Prove** that if** f(x)** is an **odd** function (antisymmetric
about the y axis; **f(-x) = -f(x)** ) then **a _{n} = 0**
for all n

Include 3 examples of *odd* functions.

5. Compare Fourier Series and Taylor Series (MA1023). When can each be used? What sorts of functions can be expanded in a Fourier Series? In a Taylor Series? Consider as a point of discussion the function f(x) defined on -1< x < +1 by f(x) = 0 if x < 0 and f(x) = 1 if x > 0 (a "step function"). Can a Taylor Series be found for it? a Fourier Series? You may want to use an outside reference book to look up conditions for each kind of series being applied.

6. Each person in your team should speak with a faculty member in the
department of your major and learn about one *application* of Fourier
Series. Include this information in your document to be turned in.

** **