Page 44 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

34 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
3.2.2 Best Approximation
We next ask whether, since we can never sum to infinity, the values of the constants c n in
Eq. (3.13) give the most accurate approximation of the function. To illustrate the idea we return
to the idea of orthogonal vectors in three-dimensional space. Suppose we want to approximate a
three-dimensional vector with a two-dimensional vector. What will be the components of the
two-dimensional vector that best approximate the three-dimensional vector?
Let the three-dimensional vector be f = c 1 1 + c 2 2 + c 3 3 . Let the two-dimensional
vector be k = a 1 1 + a 2 2 . We wish to minimize ||k − f||.

2
2

||k − f|| = (a 1 − c 1 ) + (a 2 − c 2 ) + c 2 1/2 (3.20)

3
It is clear from the above equation (and also from Fig. 3.1) that this will be minimized when
a 1 = c 1 and a 2 = c 2 .
Turning now to the orthogonal function series, we attempt to minimize the difference
between the function with an infinite number of terms and the summation only to some finite
value m. The square of the error is

b b

2 2 2 2
E = ( f (x) − K m (x)) dx = f (x) + K (x) − 2 f (x)K(x) dx (3.21)
x=a x=a
where
∞

f (x) = c n n (x) (3.22)
n=1
and
m

K m = a n n (x) (3.23)
n=1

FIGURE 3.1: Best approximation of a three-dimensional vector in two dimensions