part 1 - what is a fourier series

8/10/2019 Part 1 - What is a Fourier Series

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What is a Fourier Series?

In your calculus class, you may have studied Taylor series (if not, thats OK). Using

Taylor series, we approximate functions with polynomials using derivatives at a specified point.

Fourier series provide a function approximation that is inherently different from Taylor series;

they approximate functions using sines and cosines over an interval. Fourier series were firstused in the early 1800s by Joseph Fourier (1768-1830) to describe complicated periodic

phenomena. Since a Fourier series uses only sines and cosines, it always creates a periodic

function as the approximating function. Consequently, Fourier approximations are often appliedto the study of heat flows, oscillations, vibrations, sound, and other wave forms that exhibit

periodicity. Today, processes associated with Fourier series can be used in speech recognition,

music analysis, and in understanding how sound is affected by transmission through cell phones.

A Fourier series is an infinite trigonometric series of the form

() () () () () () ()

which can be written using summationnotation as

01

cos sink kk

F x a a k x b k x

.

Our goal in creating a Fourier series is to

approximate a given function with theFourier series given above by choosing

appropriate values for and . At right,we see the 2ndorder Fourier approximation

(Blue) to the function2xy e (Red).

Fourier Series for Even Functions

Recall that ifis an even function,() (). An even Fourier series, we will denote it

by EF x has only the cosine terms, and can be used to approximate an even function, so

0 1 2 3cos cos 2 cos 3EF x a a x a x a x . In this section, we will begin bydeveloping an even Fourier approximation for some general even function f. Later we will

expand the process to produce the general Fourier series for arbitrary functions.

Given an arbitrary even function f on the interval , , we want to find the function

EF x so that Ef x F x . This means that 0 1 2 3cos cos 2 cos 3f x a a x a x a x and, consequently,

0 1 2 3cos cos 2 cos 3f x dx a a x a x a x dx

.

1. Use the equation above to find the value of0

a in terms of f x dx


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2. Simplify cos cosnx mx nx mx using the sum and difference identities from

trigonometry and use it to evaluate cos cosnx mx dx

when m n and when m n .

3. Use the result from 2) to find the value of1

a in terms of

cos x f x dx

if

20 1 2 3cos cos cos cos cos 2 cos cos 3x f x dx a x a x a x x a x x dx

By multiplying our original function fby cosines, we can find the other coefficients.

4. Generalize to find the value of na in terms of cos nx f x dx

if

0 1 2 3cos cos cos cos cos cos 2 cos cos 3nx f x dx a nx a nx x a nx x a nx x

It might help to look at 2n and 3n first. This result gives us a rule for finding the coefficients

to approximate any even function on the interval , .

5. If 2xf x e on the interval , , use an even Fourier series and numerical integration

on your calculator to determine the coefficients 0 1 2 3 4, , , , ,a a a a a and 5 .a Compare the graph of

2xf x e to that of your series 0 1 2 5cos cos 2 cos 5EF x a a x a x a x on the

interval , .

Fourier Series for Odd Functions

Recall that ifis an odd function,() (). An odd Fourier series has only the sine terms, and

can be used to approximate an odd function, so () () () () .

1. Why is there no0

b term in the series 0F x ?

2. Using steps similar to those outlined for even functions, develop a rule for finding the coefficients to

approximate any odd function on the interval ].

3. If () ()on the interval , use an odd Fourier series and numerical integration onyour calculator to determine the coefficients and .


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General Fourier Series

Now we are ready to consider Fourier series for any function. Using steps similar to those usedabove, develop a rule for finding the coefficients to approximate an arbitrary functionfon the interval

.

In our prior work, we saw how multiplication by cos nx and integrating generates the equation

0 1 2 3cos cos cos cos cos cos 2 cos cos 3nx f x dx a nx a nx x a nx x a nx x dx

.

By evaluating the integrals, we eliminate all but one term inEF allowing us to find the value of na in

terms of the value of cos nx f x dx

. Similarly, we can eliminate all but one term in OF by

multiplying by sin nx and integrating. What we need to consider in the general form

() () () () () () ()

is how the sines and consines interact when we multiply and integrate.

1. What can you say about the value of cos sinnx kx

for all n k ?

2. Apply your technique to determine the coefficients and if 1

1 x

f xe

on the interval .

3. Compare the graphs of f and Fas you increase the number of terms used in the approximation.

part 1 - what is a fourier series

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