Partial Differential Equations PowerTool

by Dr. Jim Herod

Section 1.6: A Maximal Orthonormal Family

In the previous sections, we have developed the notions of an orthogonal family, or of an orthonormal family. We emphasized the importance of these in Fourier Analysis. There is one final notion that we introduce in this section. It is that of a maximal  orthonormal family. It should become apparent what is the importance of this notion in this section.

We review that important Fourier Inequality. Suppose that { E, <x, y> } is a linear space on which there is defined an inner product. If { , , , ... } is an orthonormal  family and f  is in the space, then

      0     =  +   - .

Several important facts follow from this inequality.

Fact 1:  Suppose that n is a positive integer, that S  is the span of { , , , ... ,  }, and that f  is in E. The closest element in S  is given by the Fourier Expansion

                                                 approximation for f =     .

We have argued that this is true in Section 1.3.

Fact 2:  If { , , , ... } is an infinite sequence, then the number series    converges.

We argue this as follows: By choosing  = < f  ,  > we get Bessel's inequality. From this we see that the number series is increasing and bounded above. That is, it is a series of non-negative numbers and bounded above. Thus, it has to converge.

Fact 3:  If { , , , ... } is an infinite sequence, then the series in E given by

converges in E.

This follows from Fact 2 through a series of steps: Suppose m > n.

          -         =   

Also,

        =   | < f ,  > |   =   | < f ,  > |   -    | < f ,  > |

Using Fact 2, we have the convergence of the vector series.

Remark: The naive novice might think that having an infinite orthonormal family and having that the series of Fact 3 converges implies that it converges to f . Only a minute or two of consideration, however, will yield the folly of this mistake. Here is a specific example. Take the orthonormal sequence to be the infinite sequence consisting of

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

Take f  to be

                               f ( x ) = .

Look at how badly the approximation misses.

>	f:=x->x^2*(1-x);

>	for n from 1 to 10 do    a[n]:=int(f(x)*sqrt(2)*sin(2*n*Pi*x),x=0..1); od;

>	approx:=x->sum(a[p]*sqrt(2)*sin(2*p*Pi*x),p=1..10);

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

>

We missed all the odd terms. There were not enough orthonormal vectors. One needs a maximal  family to be able to guarantee convergence.

Definition:  An orthogonal sequence is maximal , or complete , if the only vector in the space that is orthogonal to every element in the sequence is the zero vector.

Fact 4 . Suppose that { , , ...,  , ...} is a  maximal orthonormal sequence. Then

converges to f in norm.

We argue this by showing that the element   g  defined as  f -      is orthogonal to each . Just do the computations:

   < g ,  > = < f ,  > -  < f ,  > = 0.

Here, we have used the fact that the dot product of  and  is zero, unless p = n. In that case, the dot product is 1. Because the orthonormal sequence was maximal and g is orthogonal to every , it must be that g is zero, and the series converged to f .

We need to repair the previous example by taking all the sine terms:

                            {2 sin(  x), 2 sin(2  x), 2 sin(3  x), ... }.

>	f:=x->x^2*(1-x);

>	for n from 1 to 10 do    a[n]:=int(f(x)*sqrt(2)*sin(n*Pi*x),x=0..1); od;

>	approx:=x->sum(a[p]*sqrt(2)*sin(p*Pi*x),p=1..10);

We off set the graph of f just a little so that we can see the difference.

>	plot([f(x)+0.003,approx(x)],x=0..1,color=[black,red]);

>

This was the culmination of the review of concepts from Linear Algebra that are needed to understand some of the methods for solving partial differential equations in this set of lecture notes. The new and important idea of this Section is the notion of a maximal orthonormal family. With such a family, we can approximate in norm all vectors in the space having the associated dot product.

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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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