Mathematics 647 - Spring 2001



Special Project I - Due: March 9, 2001

Fourier Series



1. Goals

The purpose of this project is to further understand and visualize some properties about Fourier series. Some of the topics have been already discussed in class, but looking at some plots could help you further understand what is involved. The use of some computer software or graphing calculator is highly recommended for this project (and it can substantially reduce your work).


2. Rules

You are expected to hand in written solutions (typed reports and nice print out of pictures are not needed but they are very much appreciated). You are encouraged to discuss the problems with your classmates and you can do this project in a team of up to four people. If the project is done in collaboration, only one copy of the solutions needs to be hand in with all the names of the students participating in the team.


3. Problems

A. More about symmetry. We have already seen that symmetry (or parity in a function) comes very handy in computing Fourier coefficients. Here there is some more about it.

A1. Compute a few Fourier sine coefficients for each of the functions f(x)=1, g(x)=|x-1/2|, and h(x)=(x-1/2)2 on the interval (0,1). You can use a computer and list, say, the first ten coefficients.

A2. You should see a common pattern in the coefficients for these functions. Give a proof or provide some convincing arguments of why the even coefficients for the given functions are zero. Looking at the graphs of the given functions and the graphs of a few functions $\sin (n \pi x)$ should help.

B. The size of the Fourier coefficients. For a Fourier series to be convergent the coefficients should go to zero as n goes to infinity. Here we'll try to understand how fast do they go to zero.

B1. Compute a few Fourier sine coefficients for each of the functions u(x)=x, v(x)=1/2 -|x-1/2|, and w(x)=x(x-1), on the interval (0,1). Again, you can do this with a computer program.

B2. For which of the given functions do the coefficients tend to zero most rapidly as n gets larger? Look for patterns in terms of powers of n. (Some coefficients may be zero but we want to understand the size of the absolute value of the ones that are not.)

B3. What does the size of the coefficients in terms of n appear to be related to? Any conjectures about the size of the coefficients? (Hint: look at the periodic extensions of the functions.)

C. Gibbs phenomenon. Gibbs showed that near a jump discontinuity of a function f, the partial sums SN of the Fourier series of f ``overshoot'' by about 9 percent of the jump. We'll try to see this.
C1. Compute a few terms in the Fourier sine series of each of the functions D(x)=1, E(x)=x, and F(x)=x2, on the interval (0,1). Draw the plots of the corresponding partial sums or produce computer prints.

C2. What is the pointwise limit of the Fourier sine series of the given functions on (0,1)? How about x=1?

C3. Carefully examine the graphs of several partial sums for the given functions close to the point x=1. Compute for example partial sums SN with N= 5, 10, 15. Explain in your own words what you see happening close to the point x=1.

C4. In particular, look further at the partial Fourier sine sums of the function G(x)=100 on the interval (0,1). Estimate the maximum value of the partial sums SN when you use, say, N= 10, 15, or 20. (Do this by looking at the graphs; do not try to do it analytically.)

C5. Do your observations agree with Gibbs' discovery? (To see the jump discontinuity and its magnitude look at the periodic extension of the function.)