Page 304 - Matrix Analysis & Applied Linear Algebra

300 Chapter 5 Norms, Inner Products, and Orthogonality
Example 5.4.5

−1
Problem: Determine the Fourier expansion of x = 2 with respect to the
1
standard inner product and the orthonormal basis given in Example 5.4.4
      
1 1 −1
 
1 1 1
1
B = u 1 = √  −1  , u 2 = √   , u 3 = √  −1  .
 2 3 6 
0 1 2
Solution: The Fourier coefficients are
−3 2 1
ξ 1 = u 1 x = √ , ξ 2 = u 2 x = √ , ξ 3 = u 3 x = √ ,
2 3 6
so
     
−3 2 −1
1 1 1
2
x = ξ 1 u 1 + ξ 2 u 2 + ξ 3 u 3 =  3  +   +  −1  .
2 3 6
0 2 2
3
Youmay find it instructive to sketch a picture of these vectors in .
Example 5.4.6
Fourier Series. Let V be the inner-product space of real-valued functions
that are integrable on the interval (−π, π) and where the inner product and
norm are given by

π π 1/2
2
f|g = f(t)g(t)dt and f = f (t)dt .
−π −π
It’s straightforward to verify that the set of trigonometric functions

B = {1, cos t, cos 2t, . . ., sin t, sin 2t, sin 3t, . . .}
is a set of mutually orthogonal vectors, so normalizing each vector produces the
orthonormal set

!
1 cos t cos 2t sin t sin 2t sin 3t
B = √ , √ , √ ,..., √ , √ , √ ,... .
2π π π π π π
Given an arbitrary f ∈V, we construct its Fourier expansion

∞
∞
1 cos kt sin kt
F(t)= α 0 √ + α k √ + β k √ , (5.4.4)
2π π π
k=1 k=1