Page 702 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

[see Theorem 6.3.5]. An orthonormal basis for W can be obtained by applying the Gram–Schmidt process to the basis 3,
using the inner product

This yields (Exercise 6) the orthonormal basis (5)
If we introduce the notation

then on substituting 5 in 4, we obtain (6)
where (7)

In short, (8)
The numbers , , …, , , …, are called the Fourier coefficients of f.

EXAMPLE 1 Least Squares Approximations

Find the least squares approximation of on by