Math 4567 : Applied Fourier Analysis (Lecture 001)
Fall Semester 2009, 12:20 - 13:10 pm, MWF,  VinH  20

Prereq- Math 2243 or 2373 or 2573;
4 credits

Instructor: Willard Miller
Office: Vincent Hall 513 VinH
Office Hours:   01:25 P.M.-02:15 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).

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. Five homework assignments due on Fridays. About two weeks allowed for each assignment. [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, October 09, Monday November 09 & Monday, December 07 at the regular lecture time), Final Exam (December 23, 08:00-10:00)

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.

Math 4567, Lecture 01 Fall 2009  Syllabus

Week
Topic
Chapter
HW due
W SEP 09 1 Orthonormal Sets
7

F SEP 11
1 Orthonormal Sets 7

M SEP 14 2 Orthonormal Sets 7

W SEP 16 2 Introduction to  Fourier Series
1
F SEP 18 2 Introduction to  Fourier Series
1

M SEP 21
3 Introduction to  Fourier Series
1

W SEP 23
3 Convergence of  Fourier Series
2

F SEP 25 3 Convergence of  Fourier Series 2

M SEP 28 4 Convergence of  Fourier Series 2

W SEP 30
4 Convergence of  Fourier Series 2

F OCT 02
4 Convergence of  Fourier Series 2
#1
M OCT 05 5 Partial Differential Equations of Physics
3

W OCT 07 5 Partial Differential Equations of Physics 3

F OCT 09 5 Midterm 1

M OCT 12
6 Partial Differential Equations of Physics 3

W OCT 14 6 Partial Differential Equations of Physics 3

F OCT 16 6 Partial Differential Equations of Physics 3
#2
M OCT 19 7 Partial Differential Equations of Physics 3

W OCT 21
7 The Fourier Method 4

F OCT 23
7 The Fourier Method 4

M OCT 26 8 The Fourier Method 4

W OCT 28 8 Boundary value Problems 5

F OCT 30
8 Boundary value Problems 5
#3
M NOV 02
9 Boundary value Problems 5

W NOV 04
9 Sturm-Liouville Problems and Applications 8

F NOV 06 9 Sturm-Liouville Problems and Applications 8

M NOV 09 10 Midterm 2

W NOV 11
10 Sturm-Liouville Problems and Applications 8

F NOV 13
10 Sturm-Liouville Problems and Applications 8
#4
M
NOV 16
11
Sturm-Liouville Problems and Applications 8

W NOV 18 11 Sturm-Liouville Problems and Applications 8

F NOV 20
11 Sturm-Liouville Problems and Applications 8

M NOV 23
12 Sturm-Liouville Problems and Applications 8

W NOV 25 12 Fourier Integrals and Applications 6

F NOV 27 12 Thanksgiving Vacation

M NOV 30
13 Fourier Integrals and Applications 6

W DEC 02
13 Fourier Integrals and Applications 6

F DEC 04
13 Fourier Integrals and Applications 6
#5
M DEC 07 14 Midterm 3

W DEC 09 14 Fourier Integrals and Applications 6

F DEC 11
14 Fourier Integrals and Applications 6

M DEC 14
15 Fourier Integrals and Applications 6

W DEC 16 15
 Fourier Integrals and Applications

M
DEC 23

Final Exam, (last day for turning in the take home exam, my office)

Supplementary materials for the course:

Graphs of some Taylor polynomial approximations of sin(x), -4 < x < 4. Note that the Taylor polynomial T_19(x) is such a good approximation that the graphs can't be distinguished in the interval -4 < x < 4.

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)