MAT 4725 Handout 8.2 Part II
Recall (Linear Algebra p.1 – p.2)
Inner Product
Example 1
b
Let f , g  C a, b . Show that f , g   f  x  g  x  dx is an inner product on C  a, b .
a
Norm, Distance, …
Orthonormal Bases
A basis S for an inner product space V is orthonormal if
1. For u , v  S , u, v  0
2. For u  S , u is a unit vector.
1
Gram-Schmidt Process
v1 , v2 ,
, vn  Basis

w1 , w2 ,
, wn  Othogonal Basis

u1 , u2 ,
, un  Othonormal Basis
2
Definition 8.1
0 ,1, ,n  is said to be linearly independent on [a, b] if, whenever
n
 c  ( x)  0, x [a, b] ,
j 0
j
j
we have ck  0 j  0,1,..., n.
Theorem 8.2
If  j ( x ) is a polynomial of degree j , then 0 , 1 ,
interval [a, b] .
, n  is linearly independent on any
Idea
Definition
 n  the set of polynomials of degree  n
Theorem 8.3
If 0 ( x), 1 ( x),
, n ( x) is linearly independent in  n , then Q  n ,  unique c j such
n
that Q( x)   c j j ( x) .
j 0
3
Example 1
0 ( x)  2

1 ( x)  x  3

2
2 ( x)  x  2 x  7
Express Q( x)  a0  a1 x  a2 x 2 as a linear combination of 0 ( x), 1 ( x) , and 2 ( x).
4
Weight function We will use w  t   1 for the most of the examples.
Modification of the Least Squares Approximation
n
Given f  C[a, b] , approximate f ( x) by P( x)   akk ( x) .
k 0
b
Find ak such that E   w( x)  f ( x)  P( x) dx is minimized.
2
a

 b
a
w
(
x
)

(
x
)

(
x
)
dx
   w( x) f ( x) j ( x)dx for j  0,1,..., n

k 
k
j
k 0
a
 a
n
b
What if…
Definition 8.5
0 ( x),1 ( x), ,n ( x) is said to be an orthogonal set of functions on [a, b] with respect
to w if
b
jk
 0
w
(
x
)

(
x
)

(
x
)
dx


k
j
a
 j  0 j  k
(Orthonormal if all  j =1 .)
Theorem 8.6
b
 w( x) f ( x) ( x)dx
k
ak 
a
b
 w( x)  ( x)
2
k
dx

1
k
b
 w( x) f ( x) ( x)dx
k
a
a
5
Gram-Schmidt Process
0 ( x)  1
b
 x  ( x)
2
0
1 ( x)  x  B1 where B1 
dx
a
b
  ( x)
2
0
dx
a
For k  2, k ( x)   x  Bk  k 1 ( x)  Ckk  2 ( x)
b
 x k 1 ( x) dx
b
2
where Bk 
a
b
 k 1 ( x)
2
 x
k 1
and Ck 
( x)k  2 ( x)dx
a
b
 
dx
a
( x)  dx
2
k 2
a
Legendre Polynomials
P0 ( x)  1
1
 x  P ( x)
2
0
P1 ( x)  x  B1  x where B1 
dx
0
1
1
  P ( x)
2
0
dx
1
P2 ( x)   x  B2  P1 ( x)  C2 P0 ( x)  x 2 
1
3
1
 x  P1 ( x) dx
1
2
where B2 
1
1
  P ( x) 
2
1
1
dx
 xP ( x) P ( x)dx
1
 0 and C2 
0
1
1
  P ( x) 
2
0
dx

1
3
1
6
Example 2
Find the least squares approximation of f  x   sin  x  on  1,1 by the Legendre
Polynomials.
1
1
 f ( x) P ( x)dx  sin  x 1dx
0
a0 

1
1
  P ( x)
2
0
1
1
 1
2
dx
1
a1 
1
f ( x) P1 ( x)dx

1
1
  P ( x)
2
1
 sin( x)  xdx
1
 sin( x)   x
a3 
2
1
1
  dx
3
2
 2 1
1  x  3  dx
1

1
 sin( x)   x
1
3
dx
3

0
3 
 x  dx
5 
2
 3 3 
1  x  5 x  dx
1
  x

1

1
1
2
dx
1
a2 
dx
1
1

0

35  15   2 
2 3
 P( x)  a0 P0 ( x)  a1P1 ( x)  a2 P2 ( x)  a3 P3 ( x)
7