Page 699 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

9.4 In this section we shall use results about orthogonal projections in inner
product spaces to solve problems that involve approximating a given
APPROXIMATION function by simpler functions. Such problems arise in a variety of
PROBLEMS; FOURIER engineering and scientific applications.
SERIES

Best Approximations

All of the problems that we will study in this section will be special cases of the following general problem.

Approximation Problem Given a function f that is continuous on an interval , find the “best possible approximation”

to f using only functions from a specified subspace W of .

Here are some examples of such problems:

(a) Find the best possible approximation to over [0, 1] by a polynomial of the form .

(b) Find the best possible approximation to over [−1, 1] by a function of the form .

(c) Find the best possible approximation to x over by a function of the form
.

In the first example W is the subspace of spanned by 1, x, and ; in the second example W is the subspace of

spanned by 1, , , and ; and in the third example W is the subspace of spanned by 1, , ,

, and .

Measurements of Error

To solve approximation problems of the preceding types, we must make the phrase “best approximation over ”

mathematically precise; to do this, we need a precise way of measuring the error that results when one continuous function is

approximated by another over . If we were concerned only with approximating at a single point , then the error

at by an approximation would be simply

sometimes called the deviation between f and g at (Figure 9.4.1). However, we are concerned with approximation over the

entire interval , not at a single point. Consequently, in one part of the interval an approximation to f may have

smaller deviations from f than an approximation to f, and in another part of the interval it might be the other way around.

How do we decide which is the better overall approximation? What we need is some way of measuring the overall error in an

approximation . One possible measure of overall error is obtained by integrating the deviation over the

entire interval ; that is,

(1)