MATH 522

WAVELETS
Fall 2003
Professor Robert Sharpley
Meets: TTh  9:30-10:45 in LeConte College 401

Instructor Information
 Office:   LeConte College 313 D
 Office Hours: TTh 10:45-11:45 am (or by appointment)

Final Exam (updated Tuesday 12/09/03 at 7:30 pm)

The best way to prepare is to go over the previous tests. Primary topics are:
  1. Inner product spaces and orthonormal bases.
  2. Classical Fourier sine, cosine, and exponential series. The Fourier N-th partial sums.
  3. The discrete Fourier transform.
  4. The continuous and discrete Haar transforms:
    • fast implementation into scale and wavelet components.
    • the multiresolution ladder and its projections.
    • connection between the two.
  5. The Fourier transform
    • its main properties involving differentiation, translations, dilations, ...
    • band-limited functions and the Shannon sampling theorem.
    • the Gaussian; Heisenberg uncertainity inequality.
    • Fourier inversion theorem and its use.
    • Convolution.
  6. Multiresolution Analysis.
    • the scale function and its properties.
    • the corresponding wavelet.
    • examples including the Haar system, sinc functions, and Daubechies scaling functions.

Background
The subject of "Wavelets" was motivated by early orthonormal representations developed by Haar and Stromberg and inspired the development of a general theory by Yves Meyer in the mid 1980's. The fundamental mathematical discovery by Ingrid Daubechies of compactly supported, smooth wavelets in 1988 has let to an explosion of mathematical and scientific discoveries. The mathematics of wavelets has had a profound effect upon applied mathematics, image and signal processing, statistical data processing, feature identification, visualization & computer animations, and multiscale data representations. Over the last decade, the mathematical aspects of the subject have been clarified and reduced to a form now suitable for undergraduate instruction.

Course Syllabus
The course will develop the basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis. Application of the concepts will be to differential equations, data compression, signal and image processing. Numerical experiments will be used to illustrate the primary principles of multiscale decomposition, approximation, and reconstruction. Computational algorithms developed in lectures will be implemented in a high level programming language such as Matlab and/or Maple.

Prerequisites
   ( Math 544 or Math 526), or consent of department.

Required Text
   A First Course in Wavelets with Fourier Analysis, by A. Boggess and F.J. Narcowich, Prentice Hall, Upper Saddle River, NJ, 2001. [ISBN: 0-13-022809-5]

Course Grading
Two tests (25%) each, Homework (20%), and Final Exam (30%). Homework will be collected on a regular basis. Graduate Students will be required to write a term paper (counting 25% of the Final Exam) on wavelet applications in the student's subject area, as well as additional test and homework problems.

Attendance: Classroom attendance is both expected and required according to the official university 10% rule.

Important Course Dates:


Homework Assignments
Click here for an up-to-date list of Homework Assignments and other course materials.
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This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated Mar 2003. 
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