Fourier Analysis 2010

Math 4330/5330, Fourier Analysis

Semester 2, 2009-2010

Course Lecturer:

Dr. Judith Packer, Dept. of Mathematics

Tel: (303) 492-6979
Office: Math 227
Email: packer@colorado.edu
URL: http://spot.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 3001 or instructor consent. A knowledge of the rudiments of complex numbers is also required.

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. 17, 2010, 9 a.m. - 10 a.m., ECCR 139: 25 % of final grade.
Solutions to Midterm 1
Take-home mid-term exam - given out Wednesday, March 31, 2010, 10 a.m., due Wednesday April 7, 2010, 5 p.m. (please note that takehome exams will be different for Math 4330 and Math 5530): 25 % of final grade.
Click here for the take-home if you are in Math 4330!
Selected Solutions to Math 4330 Takehome Midterm 2
Click here for the take-home if you are in Math 5330!
Selected Solutions to Math 5330 Takehome Midterm 2
In-class final exam - Thursday, May 6, 2010, 7:30 a.m. - 10 a.m., in ECCR 139 - note morning hour - bring your coffee! : 30% of final grade.
Final exam from last year (2009)

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 9 - 9:50 a.m. in ECCR 139.

Office Hours:
MWF 3-4 p.m., and by appointment.

Homework:

Some Important Names associated with Fourier Analysis :

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

Fun Animations, courtesy of Dr. Eric Stade:

Section 1.6: Gibbs' phenomenon and the sawtooth function
Section 1.6: Gibbs' phenomenon in 3D (a complex-valued, piecewise smooth, discontinuous function and partial sums of its Fourier series)
Section 2.3: "Thermal imaging" of heat flow in a bar, with Neumann boundary conditions (cf. Example 2.3.1. The initial position function is f(x)=x. Red portions of the bar are warmest; blue portions are coolest. See equation (2.64) for the analytic solution)
Section 2.3: "Thermal imaging" of heat flow in a bar, with mixed boundary conditions (cf. Example 2.3.2. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.74) for the analytic solution)
Section 2.4: "Thermal imaging" of heat flow in a bar, with periodic boundary conditions (cf. Example 2.4.1. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.87) for the analytic solution)
Section 2.7: A solution to the wave equation, illustrating the principle (as reflected by the differential equation (2.149)) that acceleration is proportional to concavity
Section 2.9: D'Alembert's solution the wave equation. We assume the initial velocity g to be zero for simplicity, so that d'Alembert's formula (2.187) reads y(x,t)=(f_odd(x+ct)+f_odd(x-ct))/2. That is, we have the graph of f_odd/2 moving left with velocity c, plus the graph of f_odd/2 moving right with velocity c. To make the interaction of these two waves more apparent, we have dashed them in beyond the actual, physical boundaries x=0 and x=ell of the string itself (which appears as a solid curve)
Section 3.2: The Cau chy sequence of Figure 3.2: this sequence converges in mean square norm to zero, but does not converge pointwise on any set of real numbers.
Section 3.2.: The Cauchy sequence of Figure 3.3: (f_1 through f_22 in red, the function f to which the f_N's converge pointwise in black.)
Section 5.7: Convolution with approximate identities I. A discontinuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the top portion of Figure 5.3
Section 5.7: Convolution with approximate identities II. A continuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the bottom portion of Figure 5.3
Section 5.7: Limits of some approximate identities. The approximate identity g_epsilon of Figure 5.4, with epsilon decreasing as time evolves.

More movies:

Section 2.7: 1940 Tacoma Narrows Bridge Collapse (Quicktime), courtesy of National Archives

Back to the home page of Judith A. Packer
Last modified January 5, 2010.