Math 8370, Harmonic Analysis 1
Spring Semester 2, 2013-14
Dr. Judith Packer, Dept. of Mathematics
Tel: (303) 492-6979
Office: Math 227
Course Syllabus: For course syllabus, click here!
The material to be covered includes some of Chapter I and most of Chapters II,
III, IV, and VI in the Katznelson book. Topics to be covered include:
A review of periodic functions and T,
trigonometric polynomials, and Fourier series on the circle,
including convergence problems for Fourier series on T, including L^1 and L^2;
mean, pointwise and absolute convergence. Convolution in L^1(T). Conjugate functions,
maximal functions, and Hardy spaces. An overview of interpolation of norms and linear operators;
the Hausdorff-Young theorem. The Fourier Transform in L^1(R): convolution in
L^1(R) approximate identities and the Poisson summation formula.
The Fourier transform in L^2(R), inversion and the Plancherel Theorem.
The course will conclude with an introduction to the wavelet transform and the Haar
wavelet expansion, the multiresolution analysis construction: scaling functions and wavelets, and
if time permits, we will finish up with a discussion of Daubechies wavelets with compact support.
Math 5320,or 6320, or instructor consent.
We will use the text "An Introduction to Fourier Analysis and Wavelets" by M. Pinksy, American Math. Society, 2002 covering most of Chapters 1 - 4 and parts of Chapter 6.
We will also use as references the book "An Introduction to Harmonic Analysis" by Y. Katznelson, Dover, 1976.
Lecture Hours and Venue:
- Homework Assignments (every three weeks or so): 65 %
- Final project on topic related to
harmonic analysis, due early May: 35 % . There will be oral presentations, so please reserve the exam period
for our course, Tuesday, May 6, 1:30 p.m.-4 p.m. for presentations (both giving them and listening to them).
MWF 1 p.m.-1:50 p.m. ECCR 118
MWF 3-4 p.m., and by appointment.
Some Important Names associated with Harmonic Analysis :
- Assignment 1, due Wednesday, January 29, 2014.
- Assignment 2, due Wednesday, February 19, 2014.
- Some solutions to problems from HW 1 and HW 2.
- Assignment 3, due Wednesday, March 12, 2014.
- Assignment 4, due Friday, April 4, 2014.
- Assignment 5, due Wednesday, April 23, 2014.
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Last modified January 10, 2014.