Math
4567 : Applied Fourier
Analysis (Lecture 001)
Spring Semester 2010, 1:25  2:15 pm,
MWF, ME
102
Prereq Math 2243 or 2373 or 2573;
4 credits
Instructor: Willard
Miller
Office: Vincent Hall 513
VinH
Office Hours: 12:20
P.M.01:10 P.M. (M,F), 11:15
A.M.12:05 P.M. (W),
or by appointment
Phone: 6126247379
miller@ima.umn.edu,
miller@math.umn.edu
The Math
Library 4567 CourseLib page
Textbook: Fourier Series and Boundary Value problems, by
James Ward Brown and Ruel V. Churchill, McGrawHill, New York, 2008
(7th edition).
Paper Grader:
Shubham Debnath debna002@umn.edu
Phone:
6123015209
Class Description: This is a basic course on the representation and approximation of arbitrary functions as infinite linear combinations of simple functions, and of the applications of this idea in the physical and engineering sciences. Topics include: Orthonormal functions, best approximation in the mean, Fourier series, convergence pointwise and in the mean. Applications to boundary value problems, SturmLiouville equations, eigenfunctions, Fourier transform and applications. As time permits: Complex Fourier series and Fourier transform, FFT, Gibbs phenomena, Caesaro sums.
Policies:
Content and Style: Will cover most of Chapters 18. Homework
assignments from the textbook, and from my own notes. The theory
will predominate, but there will be considerable attention to
applications in other fields. Notes, homework assignments, examples,
practice exams, etc., will be posted online as the course develops.
Student Conduct: Statement on Scholastic Conduct: Each
student should read the college bulletin for the definitions and
possible
penalties for scholastic dishonesty. Students
suspected of cheating will be reported to the Scholastic Conduct
Committee.
Week 

Chapter 
HW due  
W  JAN 20 
1 
Introduction to Fourier Series 
1  
F  JAN 22 
1 
Introduction to Fourier Series 
1 

M  JAN 25 
2 
Introduction to Fourier Series 
1 

W  JAN 27 
2 
Convergence of Fourier Series 
2 

F  JAN 29  2 
Convergence of Fourier Series  2 
#1 
M  FEB 1 
3 
Convergence of Fourier Series  2 

W  FEB 3 
3 
Convergence of Fourier Series  2 

F  FEB 5 
3 
Convergence of Fourier Series  2 
#2 
M 
FEB 8 
4 
Convergence of Fourier Series  2 

W 
FEB 10 
4 
Convergence of Fourier Series  2 

F 
FEB 12 
4 
Convergence of Fourier Series  2 
#3 
M 
FEB 15 
5  Partial Differential Equations of Physics 
3 

W 
FEB 17 
5 
Partial Differential Equations of Physics  3 

F 
FEB 19 
5 
Midterm 1  
M 
FEB 22 
6  Partial Differential Equations of Physics  3 

W 
FEB 24 
6  Partial Differential Equations of Physics  3 

F 
FEB 26 
6 
Partial Differential Equations of Physics  3 
#4 
M 
MAR 1 
7  The Fourier Method  4 

W 
MAR 3 
7  The Fourier Method  4 

F 
MAR 5 
7 
The Fourier Method  4 

M 
MAR 8 
8  Boundary value Problems  5 

W 
MAR 10 
Midterm 2  
F 
MAR 12 
8  Boundary value Problems  5 
#5 
MAR 1519 
Spring
Break 

M 
MAR 22 
9 
Boundary Value Problems 
5 

W 
MAR 24 
9 
Boundary Value Problems 
5 

F 
MAR 26 
9 
Boundary Value Problems  5 

M 
MAR 29 
10  SturmLiouville Problems and Applications  8 

W 
MAR 31 
10  SturmLiouville Problems and Applications  8 

F 
APR 2 
10 
SturmLiouville Problems and Applications  8 
#6 
M 
APR 5 
11  SturmLiouville Problems and Applications  8 

W 
APR 7 
11  SturmLiouville Problems and Applications  8 

F 
APR 9 
11  SturmLiouville Problems and Applications  8 

M 
APR 12 
11  SturmLiouville Problems and Applications  8 

W 
APR 14 
12  Fourier Integrals and Applications  6 

F 
APR 16 
12  Fourier Integrals and Applications  6 

M 
APR 19 
13  Fourier Integrals and Applications  6 

W 
APR 21 
13  Fourier Integrals and Applications  6 

F 
APR 23 
13  Midterm 3 (take home exam to
be turned in today in class) 
#7 

M 
APR 26 
14  Fourier Integrals and Applications  6 

W 
APR 28 
14  Fourier Integrals and Applications  6 

F 
APR 30 
14  Fourier Integrals and Applications  6 

M 
MAY 3 
14  Fourier Integrals and Applications  6 

W 
MAY 5 
15 
Orthonormal Sets 
7 

F 
MAY 7 
15 
Orthonormal Sets 
7 

W 
MAY 12 
Final Exam, (last day for turning in the take home final
exam exam,
my office) 
Homework Assignment #1: Chapter 1 (page 11, problems 1, 2, 3, 4, 5).
Homework Assignment #2:
Chapter 1 (page 18, problems 4, 6), Chapter 2 (page 31, problems 3, 6),
(page 40, problems 8, 9)
Homework Assignment #3:
Chapter 2 (page 39, problems 1, 3, 5, 6, 7).
Solutions to
Homework Assignment #3
Homework Assignment #4:
Chapter 2 (page 42, problems 8). (page 54, problems 1, 5,6,7),
Chapter 3 (page 63, problem 3), (page 71, problems 1,2,8),
(page 76, problem 1).
Solutions to
Homework Assignment #4
Homework
Assignment #5 Chapter 3 (page 79, problems
1,2), (page 82, problems 1,2), (page 86, problems
2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2),
(page 102, problems 1,2,3).
Solutions to
Homework Assignment #5
Homework
Assignment #6: Chapter 5 (page 113, problem
1), (page 122, problem 1), (page 128, problem 2),
(page 133, problem 4), (page 136, problem 1). (page 146, problem 1),
Chapter 8 (page 209, problem 1)
Solutions to
Homework Assignment #6
Homework
Assignment #7: Chapter
8, (page 201, problems 1,2,3), (page 209, problems 2,
4),
(page 215, problem 3), (page 221, problem 2), (page 228,
problem 1), Chapter 6 (page 157, problem 2). (page 162, problem 1)
Solutions to Homework Assignment #7
Maple plots of examples of uniform and nonuniform convergence
Lecture
notes on Fourier series (PDF file) These are taken from
material (copyright by Steve Damelin and Willard Miller) for a more
advanced course. They contain detailed information about Gibbs
phenomena, Cesàro sums and other topics.
Lecture
notes on the Fourier transform (PDF file)
These are taken from material (copyright by Steve Damelin and Willard
Miller) for a more advanced course.
Lecture Notes and Background Materials on Linear Operators in Hilbert Space (pdf file) ( postscript file)