Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 271: Applied and Computational Harmonic Analysis
Approved: 2009-11-01, Naoki Saito and Thomas Strohmer

Units/Lecture:

Winter, alternate years; 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
No required textbook. Instructor will supply info about some key articles to read during the course. See below for optional reference books.

Prerequisites:

(MAT 125B or MAT 201C) and (MAT 128B or MAT 167) and (MAT 129 or equivalent), or consent of instructor.

Course Description:

Introduction to mathematical basic building blocks (wavelets, local Fourier basis, and their relatives) useful for diverse fields (signal and image processing, numerical analysis, and statistics). Emphasis on the connection between the continuum and the discrete worlds.

Suggested Schedule:

Lectures Sections Topics/Comments


Fourier series and transforms, the Shannon sampling theorem


Discrete Fourier, cosine, and sine transforms


Karhunen-Loeve transform and principal component analysis


Frame theory


Frames and sparse representations


The uncertainty principle


Windowed (or short-time) Fourier transform


Gabor (Weyl-Heisenberg) systems


Continuous and Discrete wavelet algorithms


Multiresolution analysis and fast wavelet algorithms


Applications to signal/image processing and communication

Additional Notes:

Optional reference books:
  • S. Mallat, A Wavelet Tour of Signal Processing, 3rd Ed., Academic Press, 2009.
  • I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
  • S. Jaffard, Y. Meyer, and R. Ryan, Wavelets: Tools for Science and Technology, SIAM, 2001.
  • K. Groechenig, Foundations of Time-Frequency Analysis, Birkhauser, 2001.
  • W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform, SIAM, 1995.