Homework Solutions For Cal III+

Orthogonal Functions

by Mohammad Ghomi (The T.A.)

Problem #1

You do not need to use Mathematica to find out for which real numbers a the function x^a is in  L^2(0,1). But you could check your answer for certain values. For instance suppose

then we can easily check that

Hence,  for a=0.5, x^a is in L^2(0,1). Note that the answer is only an approximation. You can check other values by changing alpha.

Problem #2
To see whether log(x) is in L^2(0,1), we first need to integrate (log(x))^2. This can be done by integrating by parts ( Let u=log(x), and dv=log(x)dx). Using Mathematica, we can easily verify that the answer is:

Next, we  need to evaluate the above integral from 0 to 1. This involves a limit, because log(x) is not defined at x=0. Again, using Mathematica, we verify that the answer is:

Hence the answer to the above question is affirmative, because the integral is finite.

Problem #3

If the Legendre polynomials p_n(x) are defined as the n^th coefficient of the Taylor series around zero of the following function:

then we can easily compute them:

The "normalized" Legendre polynomials are defined by multimplying the n^th term in the above series by ((2n+1)/2)^(1/2).

Perhaps the Legendre polynomials may be defined more naturally as the orthogonal basis which is obtained by applying the Gram-Schmidt  process to the elements {1,  x,  x^2,  x^3, ...}.

Using Mathematica, we can verify that the first n normalized Legendre polynomials coincide with the n elements obtained by orthonormalizing {1, x,  x^2, ..., x^n}. First, we need to load the apropriate package:

Now we can easily get the first four terms as follows:

So,  as you can see, these terms are normalization of the coefficients we obtained above.

Problem #4

To see what the graphs for the partial sums of the series sin nx/n are going to look like, start by setting:

This defines the n^th partial sum. The n^th graph, is generated by the command

Now we can generate the first five graphs as follows:

Now it is easy to see what the limit looks like. For good measure throw in a high number, say 100:

As you would suspect, the limit is some kind of a sawtooth.