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MATH262-10S2

EMTH202-10WASSIGNMENT 3 – 2010

This assignment is due at 4:00 PM Wednesday 13 October and is to be handed through the boxes onLevel 4 in the Mathematics and Computer Science Building. For MATH262 students, this assignment

is worth 10% of your final grade. For EMTH202 students it is worth 5% of the final grade. You maywork by yourself or with one other person. If you hand in a joint assignment, you will each be given thesame mark.The assignment should include any appropriate output from MAPLE. You should explain what you aredoing and why. As alawyas, your comments should be in English!

The Background

In the lectures we saw that, if a function f and its derivative f are piecewise continuous, then the Fourierseries will converge pointwise at points where f is continuous and to 1

2(f(x+)+f(x−)) at all points where

f is not continuous. In this assignment we will explore in more detail the nature of this convergence.

The Investigation

1. Find the Fourier series for

f(x) =

1 for 0 < x < π 

0 for x = − π , 0, π 

− 1 for − π < x < 0

with f(x + 2π ) = f(x).

By the convergence theorem, the Fourier series will converge to f(x) for all  x (this is the reason we have

given the value 0 to each of the points of discontinuity). Let

sN(x) =4

π 

N−1n=0

sin (2n + 1)x

2n + 1;

the sum of the first N non-zero real terms.

2. Plot (on the same set of axes) f(x), s4(x), s10(x) and s100(x). Produce another plot that showsa neighbourhood of the region to the right of 0 in some detail. What do you notice?

3. Compute

 π

−π sN(x) dx.

What do you notice? (Hint: Compare withπ

−πf(x) dx.)

4. Show that

s

N(x) =

2

π 

sin 2Nx

sin xx = lπ 

 4N

π (−1)l x = lπ 

where l is any integer.

5. Show that the first maximum on R+ of  sN(x) occurs at

ξN =π 

2N.

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6. Plot the error in the approximation of  f(x) by sN(x) at ξN as a function of  N from N = 1 toN = 20. What do you notice?

7. Show that

limN→

∞sN(ξN) =

2

π 

 π

0

sin t

tdt.

Use MAPLE to evaluate this integral. What does this say about the error of approximation of  f(x)

by sN(x)?

What does it all mean?

In this investigation, we have seen that

limN→∞ sN(x) = f(x)

butlimN→∞

sN(ξN) = 1.

What this says is that while sN(x) does approximate to f(x), the error in the approximation does not

go to 0 as N becomes large. The location of the maximum error ξN does approach 0. Thus, in aneighbourhood of zero, we will always have the same maximum error no matter how many terms we usein the series! However, away from discontinuities, the error in the approximation will tend to 0 as N

increases. In other works, the convergence is not uniform.For a continuous function (with piecewise continuous derivative), the Fourier series converges bothpointwise  and uniformly . However, as this example shows, a function with jump discontinuities willconverge pointwise  but not uniformly . By contrast, approximation of a function by its Taylor series isuniform within its radius of convergence.This example is not special. Any function which has a jump discontinuity will display the same property.

This behaviour is called the Gibbs Phenomenon. The “overshoot” is roughly 8.95% of the jump heightfor any  such function. Thus, using sN(x) to approximate such functions, we will always have an error of around 9% for some value of  x irrespective of the number of terms used in the approximation!