1 Math 472/572 - Fourier analysis and wavelets

### MATH 472/572 - FOURIER ANALYSIS AND WAVELETSFall 2001

This class is cross-listed as:

• Math 472 - Call # 17520 - Fourier analysis and wavelets
• Math 572 - Call # 20135 - Fourier analysis and wavelets

This course is an introduction to Fourier Analysis and Wavelets. It has been specifically designed for engineers, scientists, and mathematicians interested in the basic ideas underlying Fourier analysis, wavelets and their applications.
This course integrates the classical Fourier Theory with its latest offspring, the Theory of Wavelets. Wavelets and Fourier analysis are invaluable tools for researchers in many areas of mathematics and the applied sciences, to name a few: signal processing, statistics, physics, differential equations, numerical analysis, geophysics, medical imaging, fractals, harmonic analysis, etc. It is their multidisciplinary nature that makes these theories so appealing.

Topics will include:

• Discrete Fourier Transform (including Fast Fourier Transform). Circular convolution, translation invariant linear transformations. Discrete wavelets (including Fast Wavelet Transform via filter banks).
• Fourier series and wavelets in sequence space. Orthogonal expansions on the interval: trigonometric functions, Haar basis, orthogonal polynomials. Sampling theorems.
• Fourier integrals. Orthogonal expansions on the line: Haar basis. Time/frequency analysis, windowed Fourier Transform, Gabor basis. Multiresolution analysis on the line. Wavelets. Basic examples: Meyer's, Shannon's and Daubechies' compactly supported wavelets. Biorthogonal wavelets, wavelet packets, wavelets on the interval.
• Applications to signal processing: data compression, storage and retrieval problems, image processing, pattern and speech recognition; and to statistics: denoising, linear and non-linear approximation, treshholding.

Numerical experiments are necessary to fully understand the scope of the theory. The use of some Wavelet Toolbox will be encouraged. There exists a WAVELAB package which is Matlab/Octave based software designed by a team at Stanford and available for free on the Internet. Another software available for free is KHOROS which has been designed at EECE/UNM and Los Alamos. MATLAB, in particular the Wavelets Toolbox, is available on the University computer system. MATLAB 6 is available in the Mathematics and Statistics Department Computer Laboratory.

Texts:

• An Introduction to Wavelets Through Linear Algebra by Michael Frazier. Springer Verlag, Feb 1999; \$49.95; ISBN: 0387986391.
• Fourier series and integrals by H. Dym and H.P. McKean. Academic Press, 1986, \$59; ISBN: 0122264517
• A Wavelet Tour of Signal Processing by S. Mallat, Second Edition, Academic Press, 1999, \$59.95; SBN: 12466606X