## Math 4330/5330, Fourier Analysis

## Semester 2, 2018-19

**Course Lecturer: **
#### Dr. Judith Packer, Dept. of Mathematics

*Tel: (303) 492-6979*

Office: Math 227

Email: packer@colorado.edu

URL: http://math.colorado.edu/~packer

**Course Information:**

Fourier analysis, first developed by Joseph Fourier in the 1800's, is a way of studying functions by decomposing them into certain
types of "building block" functions. In calculus, you have learned how nice enough functions can be given Taylor series
expansions and approximated by polynomials. Fourier's idea was that nice enough functions on closed, bounded intervals of
**R** could be given a infinite series expansion involving the trigonometric functions
{cosnx: n= 0, 1, 2, ...}, and {sin nx: n = 1, 2, ...}. This great idea was very fruitful and had many applications,
particularly in the field of partial differential equations coming from physics.

The material to be covered includes the Appendix, most of Chapters 1 and 3, and parts of
Chapters 2, 5, and 6, in the book "Fourier Analysis"
by Eric Stade. Topics include:

A review of the definition and arithmetic of complex numbers, periodic functions on the real line
**R**, and functions on the circle **T**, trigonometric functions with period 2 pi,
Fourier coefficients of periodic functions, Fourier series, convergence of Fourier series,
Gibb's phenonomenon for Fourier series at points of discontinuity, study of uniform convergence, differentiaion and
integration of Fourier series, Fourier series with other periods,
applications of Fourier series to solutions to boundary value problems in partial differntial equations: the heat equation and the wave equation,
vector spaces of functions, L^2 spaces and inner products, the Hilbert space L^2(**[-pi, pi],**), orthogonality and orthonormal
bases for L^2(**[-pi, pi],**), functions defined on **R**, the function space L^1(**R**), convolution of functions defined on **R**, the Fourier transform on L^1(**R**),
Fourier inversion in L^1(**R**), the Hilbert space L^2(**R**), the Fourier transform and Fourier inversion in L^2(**R**).

**Prerequisite:**

Math 2400 Calc. 3 and Math 2130 Linear Algebra. A knowledge of the rudiments of complex numbers is also required,
and Math 3001 Introduction to Analysis would be helpful background.

**Course Text:**

We will use the text "Fourier Analysis" by Eric Stade, J. Wiley and Sons, 2005, covering the Appendix,
most of Chapter 1, parts of Chapter 2, most of Chapter 3, and parts of Chapters 5 and 6.

**Assessment:**
- Homework will be assigned every week. Some, but not all, of the problems will be graded. Please note that the assigments
for Math 5330 will include extra problems. The assessment of homework performance will count for 20% of the final grade.
- In-class mid-term exam - Wednesday, Feb. 13, 2019, 1 p.m. - 1:50 p.m., ECCR 108: 25 % of final grade.
- Sample midterm from 2014 - click HERE for the sample midterm.
- Solutions to Midterm 1 - click HERE for solutions to the Feb. 13 midterm.
- Take-home mid-term exam - given out Wednesday, April 3, 2019, 1:50 p.m., due Wednesday April 10, 2019,
5 p.m. (please note that takehome exams will be different for Math 4330 and Math 5530): 25 % of final grade.
- Click HERE for Math 4330 takehome.
- Click for some solutions to the Math 4330 takehome.
- Click HERE for Math 5330 takehome.
- Click HERE for some solutions to the Math 5330 takehome.
- In-class final exam - Monday, May 6, 1:30 p.m. - 4 p.m., ECCR 108: 30% of final grade.
- Click HERE for a final exam from several years ago.
- Click HERE for some handwritten solutions to the exam above.
- Click HERE for a detailed syllabus of the course, including all university
policies and procedures.

If you are absent from an exam, or do not hand in the take-home exam on time, without a valid excuse, you will receive a
grade of "F" for that exam. Examples of valid excuse are: documented illness
(doctor's letter required), religious observance, and serious family emergency.

**Lecture Hours and Venue: **

MWF 1 - 1:50 p.m. in ECCR 108.

**Office Hours: **

M 4 p.m. - 5 p.m., WF 3 - 4 p.m., and by appointment.

** Homework: **

- Assignment 1 ,due Wed. January 23, 2019.
- Some solutions to HW 1.
- Assignment 2 , due Monday January 28, 2019.
- Some solutions to HW 2.
- Assignment 3 , due Monday February 4,2019.
- Some solutions to HW 3.
- Assignment 4 , due Monday, February 11, 2019.
- Some solutions to HW 4.
- Assignment 5 , due Monday, February 25, 2019.
- Some solutions to HW 5.
- Assignment 6 , due Monday, March 4, 2019.
- Some solutions to HW 6.
- Assignment 7 , due Monday, March 11, 2019.
- Some solutions to HW 7.
- Assignment 8 , due Wednesday, March 20, 2019.
- Some solutions to HW 8.
- Assignment 9 , due Wednesday April 3, 2019.
- Some solutions to HW 9.
- Assignment 10 , due Wednesday, April 17, 2019.
- Some solutions to HW 10.
- Assignment 11, due Wednesday, April 24, 2019.
- Some solutions to HW 11.

**Some Important Names associated with Fourier Analysis :**

** Fun Animations (Quicktime), courtesy of Dr. Alfred Clark Jr., University of Rochester: **
** More movies: **

Back to the home page of Judith A. Packer

Last modified January 14, 2019.