Math 595 HA, Fall 2007

Math 595 HA: Frames and Riesz Bases in Harmonic Analysis

Instructor: Richard Laugesen
e-mail: Laugesen@uiuc.edu
Office: 376 Altgeld Hall Phone: 333-1329
Office hours: Mon 4-5pm (Altgeld 376), Wed 4-5pm (Homework Study Session, Altgeld 443), Thu 5-6pm (Altgeld 376), or by email appointment.
Class: MWF 2-3pm in 148 Henry Bldg, for the second half of the semester (15 October - 7 December)
Text: Frames and Bases in Mathematics and Engineering: an Introductory Course by O. Christensen (not yet published; buy it as a coursepack).

Mexican hat wavelet frame generators

Course outline

Frames are special "overcomplete bases" in a Hilbert space. They arose originally in the study of nonharmonic Fourier series, and have since found many uses in harmonic analysis and signal processing.
We will first study synthesis, analysis and frames for the finite dimensional case in Rⁿ, and develop the infinite dimensional theory in Hilbert space. Then we investigate two important kinds of frame from harmonic analysis: Gabor frames (time-frequency shifts of a window function) and Wavelet frames (time shifts/scale dilates of a mother wavelet).

Chapter 1 - Frames in finite-dimensional inner product spaces
Chapter 2 - Inifinite dimensional vector spaces and sequences
Chapter 3 - Bases
Chapter 5 - Frames in Hilbert spaces
Chapter 7 - Frames of translates
Chapter 8,9,10 - Gabor frames in L²(R) and l²(Z), and applications in Signal Processing
Chapter 11 - Wavelet frames, and applications in Signal Processing

The course grade will be based on the Final Project. Here are the project handouts prepared by students:

Introduction to Finite Normalized Tight Frames by Jiahao Chen and Inmi Kim
A Brief Introduction to Grassmannian Frames by Arvind Ganesh
The Projection Method for Finite-Dimensional Approximation of the Inverse Frame Operator by Merc Chasman and Jeff Snyder
Application of Frames to Erasure Channels by Nitin Aggarwal and Kiryung Lee
Frames in CDMA Communication Systems: Tight Frames and Their Fundamental Inequality by Ka Lung Law and Behzad Sharif

Prerequisites

The course is aimed at applied mathematics and engineering students. The prerequisites are knowledge of the basics of Fourier transforms and Hilbert space theory (orthonormal bases, bounded linear functionals, and bounded linear operators from one Hilbert space to another).

Please contact me if you would like to discuss the course or its prerequisites. - Richard Laugesen Laugesen@uiuc.edu