Mathematics 248: Topics in Analysis (Wavelets) (Fall 2003)
Instructor
Svetlana Roudenko
Description
A recent noteworthy area of focus in Fourier analysis is orthogonal expansions
in a wavelet basis. The theory of wavelets is a very active area of
research with many real world applications.
This course will be an introduction to the theory of Fourier analysis,
wavelets, and related topics. More precisely, the tentative topics are
- general properties of orthogonal systems, Riesz basis, frames
- convergence and summability of Fourier series
- Fourier transforms of L2 functions, inversion and Plancherel's theorems
- Poisson summation formula
- parts of distribution theory, including Fourier transforms of tempered distributions
- general theory of wavelets (linear algebra construction and multi-resolutional analysis)
- examples of Meyer, Daubechies wavelet bases
- wavelets, wavelet-like transforms and Calderon-Zygmund operators in various function spaces
- parts of Littlewood-Paley theory
- applications to signal and image processing and to solutions of differential equations
Prerequisites
Basic analysis (Math 203 and 204) and linear algebra
Texts
- Wojtaszczyk, "Mathematical Introduction to Wavelets"
- Duoandikoetxea, "Fourier Analysis"
- Frazier, "An introduction to wavelets through linear algebra"
- Hernandez and Weiss, "A first course on Wavelets"
Course Website
For more information see www.math.duke.edu/~svetlana/math248.html
Grading
There will be several homework assignments,
MATLAB assignments and project(s) (to be presented in class)
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