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(Part 1: Linear Algebra Concepts Part 2: Orthogonality of FCNS, Intro To Fourier Series) (Haberman Sect 5.5 App) (Continuing From Last Time... )

This document summarizes key concepts from a lecture on linear algebra and Fourier series. Part 1 discusses eigenvector expansions of vectors and solutions to linear systems using complete sets of eigenvectors. If the matrix L is self-adjoint, the eigenvalues are real and the eigenvectors form an orthogonal basis. Part 2 introduces Fourier series expansions of functions on an interval. Inner products of functions are defined using a weight function. If the basis functions are eigenfunctions of a self-adjoint operator, they form an orthogonal basis and Fourier coefficients can be determined from inner products with the basis functions.

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142 views6 pages

(Part 1: Linear Algebra Concepts Part 2: Orthogonality of FCNS, Intro To Fourier Series) (Haberman Sect 5.5 App) (Continuing From Last Time... )

This document summarizes key concepts from a lecture on linear algebra and Fourier series. Part 1 discusses eigenvector expansions of vectors and solutions to linear systems using complete sets of eigenvectors. If the matrix L is self-adjoint, the eigenvalues are real and the eigenvectors form an orthogonal basis. Part 2 introduces Fourier series expansions of functions on an interval. Inner products of functions are defined using a weight function. If the basis functions are eigenfunctions of a self-adjoint operator, they form an orthogonal basis and Fourier coefficients can be determined from inner products with the basis functions.

Uploaded by

Alexander Bennett
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Math 551, Duke University Lecture 2

(Part 1: Linear Algebra concepts Part 2: Orthogonality of fcns, intro to Fourier Series)
Part 1 : [Haberman Sect 5.5 App]
Extending concepts from Linear Algebra (conclusion) (continuing from last time...)

If you have a real matrix Lnn that has a complete set of n eigenvectors,
and you use the definition of the inner product as hu, vi = uT v then:
The adjoint is L = LT
The eigenvalues can be found from the determinant, and k = k
The eigenvectors k and adjoint eigenvectors k can be found via algebra
Bi-orthogonality of eigenvectors: hk , j i = 0 if k 6= j
and otherwise hk , k i =
6 0 (both from k-th eigenmode)
Then any given w Rn can be written as an eigen-expansion form:
n
X hw, k i
w= ck k ck =
k=1
hk , k i
the set of k s form a complete basis set for expansions of any vector w. This
property guarantees that problems for w can be solved using the expansion.
Solving linear equations : Lu = b for unknown u
The eigenvector expansion method: start with the expansion formula for u:
n
X h k , ui
u= ck k ck =
k=1
h k , k i
But now u is NOT known, so cant work out numerator inner products in ck ...
So, indirect approach: go back to the original problem
Do the orthogonal projection of Eqn onto each k for k = 1, 2, , n

h k , Lui = h k , bi hk , Eqni
hL k , ui = (adjoint relation)
hk k , ui = (adjoint eig-prob, L = )
k h k , ui = (linearity and k = k )
k h k , ui = (numerator in coeff ck !)
k ck h k , k i = h k , bi (using h k , ui = ck h k , k i)

n
h k , bi X h k , bi
ck = u= k
k h k , k i k=1
k h k , k i
Self-Adjoint problems : an important special case

If L = L (symmetric real matrices, LT = L) then some results simplify:


All Eigenvalues k = k (unchanged)
All Eigenvectors k = k (only need to find 1 set of vectors!)
The coefficients in the expansion for a given vector w simplify to
n
hw, k i X
ck = w= ck k
|k |2 k=1

The coefficients in the expansion for the solution of Lu = b simplify to


n
hk , bi X
ck = u= ck k
k |k |2 k=1

Eigenvalues k are all real numbers (see HW)


The set of Eigenvectors is self-orthogonal: j k for j 6= k (see HW)

The self-adjoint version of the vector Eigen-expansion of carries over directly to


yield Fourier series for expansions of functions...
Part 2 : Fourier Series and Orthogonal-expansions of functions
Goals:
(I) To express complicated functions as sums of simple basis functions, as
Generalized Fourier series expansions
X
f (x)= ck k (x) on a x b
k

(II) To express solutions of differential equations (DE) problems as Fourier series


and reduce DE problems to simpler algebra for the ck coefficients.

Inner products for real-valued functions on an interval a x b (Definition)


Z b
hf, gi [f (x)g(x)](x) dx
a

(x) 0: positive weight function (weighted inner product )


Z b
Generalization for complex-valued fcns: hf, gi [f (x)g(x)](x) dx
a
Specifying (x) and a, b defines the inner product for a problem.
The uniform weight case: (x) 1 (classic/default case)
Z b
hf, gi = [f (x)g(x)] dx standard L2 inner product
a
s 2
Z b
2
The L norm: (||f ||2 )2 = hf, f i = f 2 dx 0
a

L2 functions: also called square integrable fcns, have finite L2 norm:

||f ||2 <

L2 fcns can blow-up as long they arent too badly behaved. Examples:
(a) f (x) = x1/4 on 0 x 1: f (0) but
Z 1 1
2 1/4 2 1/2

||f ||2 = (x ) dx = 2x = 2 Ok, L2 fcn
0 0

(b) f (x) = x1/2 on 0 x 1: f (0) and


Z 1 1
2 1/2 2

||f ||2 = (x ) dx = ln(x) = NOT L2
0 0

Fourier series theory and eigen-expansions are guaranteed to work for L2 fcns
Orthogonality of functions on interval a x b is defined using the
-weighted inner product integral hf, gi = 0

Assume {k (x)} is a complete set of basis functions


General orthogonal projection works as usual for the expansion of fcns:

X hk (x), f (x)i
f (x)= ck k (x) ck =
k
hk (x), k (x)i

Assume that the k (x)s are the eigenfunctions of a self-adjoint linear


operator (L = L), so k = k and the set of k s is self-orthogonal

(general case) hj , k i = 0 for j 6= k bi-orthogonal

(self-adjoint case) hj , k i = 0 for j 6= k self-orthogonal

hk (x), f (x)i
For the self-adjoint (Fourier series) case we have ck =
||k (x)||2
IOUs: (1) what does = mean? and
(2) what is the self-adjoint operator L? (later)

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