Partial Differential Equations PowerTool

by Dr. Jim Herod

Section 1.5: Projections

By now, we have become familiar with the tools needed to discuss Fourier Series. They are the tools of Linear Algebra. We want a linear space E of vectors. These vectors may themselves be functions on an interval. We want there to be defined on this linear space E a dot product  < x , y >. With these tools - a linear space and a dot product - we can discuss orthogonality and best approximations. There are two more topics that are useful concepts in thinking of Fourier Series. We deal with these last two topics from Linear Algebra in this section and the next.

In this section, we consider projections P . A projection on a linear space is a linear function having the property that

                                  = P.

We make closest point projections in this Section. To do this, suppose we have an orthogonal sequence of elements. We take O  to be a finite subset of these orthogonal elements:

                               O = { , ... ,  }.

Let S  be the span of the 's. That is,

                    S  = { u : u =  ,  a real number}.

We have seen that if f  is in E, then

is in S  and is closer to f  than any other element in S . Choosing this closest point defines a projection, for if

                  P(f)  =   

then   . That is, take f . Form the Fourier Expansion of f using the orthogonal vectors in O . Call this P(f) . Form the Fourier expansion of P(f) . What you get is P(f) , again.

It is interesting to take a plane in  and to form the closest point projection onto the plane. This is properly a problem for a linear algebra set of lecture notes, and we resist the temptation to do this here. What we would do to do that problem is to find two linearly independent vectors in the plane, perform the Gramm-Schmidt process on those two vectors, and construct the closest point projection as suggested above.

Instead of doing that interesting problem, we take the space C([0, 1]), and the subspace S  to be the span of

             { sin(  x), sin(2  x), sin(3  x), sin(4  x), sin(5  x), sin(6  x)}.

The vectors sin(  x), sin(2  x), sin(3  x), sin(4  x), sin(5  x),and sin(6  x)  are already orthogonal in that subspace. Now, we can make the closest point projection on S . We illustrate by getting the closest point projection onto the function

                        f(x) = Heaviside(x-1/2).

First, we draw a graph of this function.

>	f:=x->Heaviside(x-1/2);

>	plot(f(x),x=0..1);

>

Next, we compute the Fourier Coefficients.

>	for n from 1 to 6 do    a[n]:=int(f(x)*sin(n*Pi*x),x=0..1)/int(sin(n*Pi*x)^2,x=0..1); od;

>

Finally, we form the Fourier Sum as the projection onto S  and plot both f  and its approximation on the same graph.

>	approx:=x->sum(a[p]*sin(p*Pi*x),p=1..6);

>	plot([f(x),approx(x)],x=0..1);

>

Not so bad. Should it be that you think the approximation is not so good, go back and take more terms. Change the 6 terms to 10, or 20, or whatever.

Before ending this discussion, there is one check we should make. Is ? That is, are the Fourier coefficients for the approximation the same as they were for f ?

>	for n from 1 to 6 do    b[n]:=int(approx(x)*sin(n*Pi*x),x=0..1)/                  int(sin(n*Pi*x)^2,x=0..1); od;

We compare to see if the coefficients for f  and for its approximation are the same.

>	for n from 1 to 6 do    is(b[n]=a[n]); od;

>

>

The implication is that .

In this Section, we have shown that the closest point approximation is a projection of the entire space onto the span of the orthogonal elements used.

EMAIL: herod@math.gatech.edu   or    jherod@tds.net

URL: http://www.math.gatech.edu/~herod

Copyright ©  2003  by James V. Herod

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