Math 595 HA: Frames and Riesz Bases in Harmonic Analysis 


Mexican hat wavelet frame generators 
Course outlineFrames 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 R^{n}, and develop the infinite dimensional theory in Hilbert space. Then we investigate two important kinds of frame from harmonic analysis: Gabor frames (timefrequency shifts of a window function) and Wavelet frames (time shifts/scale dilates of a mother wavelet).
The course grade will be based on the Final Project. Here are the project handouts prepared by students:
PrerequisitesThe 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@illinois.edu 