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: 612-624-7379
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, McGraw-Hill, New York, 2008 (7th edition).

Paper Grader: Shubham Debnath debna002@umn.edu
Phone: 612-301-5209

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, Sturm-Liouville equations, eigenfunctions, Fourier transform and applications. As time permits: Complex Fourier series and Fourier transform, FFT, Gibbs phenomena, Caesaro sums.

Policies:

  1. Seven homework assignments due on Fridays. About two weeks allowed for each assignment that is due after February 12. [NO late homework will be accepted without good excuses]. It is fine to do homework in collaboration with your classmates, but your writeup should be your own work.
  2. Three Midterms plus the Final Exam [NO makeup tests without rigorous emergency reasons. Athletes please present "Proofs of Activities" in advance].
  3. Grading Policy (percent of total grade): Homework (20%), Three Midterms (15%+17.5%+17.5%), Final Exam (30%)
  4. Grades will be determined  through an interaction of objective standards and experience with the class. I don't preassign the number of students who will receive a specific grade. On the other hand, neither will I preassign the gradelines before seeing the distribution of grades. Gradelines will be announced on the web, as soon as possible after the quiz or exam.
  5.  Incompletes will be given only in cases where the student has completed all but a small fraction of the course with a grade of C or better and a severe unexpected event prevents completion of the course. In particular, if you get behind, you cannot ``bail out'' by taking an incomplete. The last day to cancel, without permission from your College office, is the last day of the sixth week.
Exam dates: Midterms (Friday, February 19, Wednesday, March 10 & Friday, April 23 at the regular lecture time), Final take home Exam (due Wednesday, May 12)

Content and Style: Will cover most of Chapters 1-8. 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.


Midterm 1, February 19. There are a total of 100 points on this 55 minute exam.  To get full credit for a problem you must show the details of your work. Answers unsupported by  an argument will get little credit. A standard calculator and ONE 8.5 by 11 inch sheet of notes are allowed, but no books, other notes, cell phones or other elecronic devices are allowed. Do all of your calculations on the test paper.

Midterm 2, March 10 . There are a total of 100 points on this 55 minute exam.  To get full credit for a problem you must show the details of your work. Answers unsupported by  an argument will get little credit. A standard calculator and TWO 8.5 by 11 inch sheets of notes are allowed, but no books, other notes, cell phones or other elecronic devices are allowed. Do all of your calculations on the test paper.

Midterm 3,  take home, due April 23. There are a total of 100 points on this  exam. 

Final Exam,  take home, due May 12. There are a total of 180 points on this  exam. 

Math 4567, Lecture 01 Spring 2010 Syllabus


Week
Topic
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 15-19

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 Sturm-Liouville Problems and Applications 8

W
MAR 31
10 Sturm-Liouville Problems and Applications 8

F
APR 2
10
Sturm-Liouville Problems and Applications 8
#6
M
APR 5
11 Sturm-Liouville Problems and Applications 8

W
APR 7
11 Sturm-Liouville Problems and Applications 8

F
APR 9
11 Sturm-Liouville Problems and Applications 8

M
APR 12
11 Sturm-Liouville 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)


Solutions to Midterm 1


Solutions to Midterm 2

Solutions to Midterm 3


Homework assignments:

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



Supplementary materials for the course:

A note on four types of convergence (Postscript file)  (PDF file)

The Mean Value Theorem, Extended Mean Value Theorem and L'Hospital's Rule

Maple plots of examples of uniform and non-uniform convergence

Maple plots for Fourier series, demonstrating the behavior of the kernel function D_k(t), Gibbs phenomena, the sinc function, and Cesàro sums

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)