Page 703 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

1. a trigonometric polynomial of order 2 or less; (9a)
2. a trigonometric polynomial of order n or less. (9b)
(9c)
Solution (a)

For , integration by parts yields (verify)

Thus the least squares approximation to x on by a trigonometric polynomial of order 2 or less is
or, from 9a, 9b, and 9c,

Solution (b) by a trigonometric polynomial of order n or less is

The least squares approximation to x on

or, from (9a), (9b), and(9c),

The graphs of and some of these approximations are shown in Figure 9.4.4.

Figure 9.4.4
It is natural to expect that the mean square error will diminish as the number of terms in the least squares approximation